LONDON MATHEMATICAL SOCIETY
MONOGRAPHS NEW SERIES
Series Editors
P. M. Cohn H. G. Dales
LONDON MATHEMATICAL SOCIETY MONOGRAPHS
NEW SERIES
Previous volumes of the LMS Monographs were published by Academic Press,
to whom all enquiries should be addressed. Volumes in the New Series will be
published by Oxford University Press throughout the world.
NEW SERIES
1.	Diophantine inequalities R. C. Baker
2.	The Schur multiplier Gregory Karpilovsky
3.	Existentially closed groups Graham Higman and Elizabeth Scott
4.	The asymptotic solution of linear differential systems M. S. P. Eastham
5.	The restricted Burnside problem Michael Vaughan-Lee
6.	Pluripotential theory Maciej Klimek
7.	Free Lie algebras Christophe Reutenauer
Free Lie Algebras
Christophe Reutenauer
Univer site du Quebec d Montreal
CLARENDON PRESS OXFORD
1993
Oxford University Press, Walton Street, Oxford 0X2 6DP
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© Christophe Reutenauer, 1993
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Library of Congress Cataloging in Publication Data
Reutenauer, Christophe.
Free lie algebras/Christophe Reutenauer.
(London Mathematical Society monographs, new
series, no. 7)
Includes bibliographical references and index.
1. Lie algebras. I. Title. IL Series.
QA252.3.R48	1993	512'.55-dc20	92-27318
ISBN 0-19-853679-8
Typeset by Integral Typesetting, Great Yarmouth, Norfolk
Printed in Great Britain by St Edmundsbury Press,
Bury St Edmunds, Suffolk
Ce livre est dedie d
Arthur, Victor, Emile, Eva

Preface
Lie polynomials first appeared at the turn of the century in the work of
Campbell, Baker, and Hausdorff on exponential mapping in a Lie group,
which led to a result known as the Campbell-Baker-Hausdorff formula.
About thirty years later, Witt showed that the Lie algebra of Lie polynomials
is actually the free Lie algebra, and that its enveloping algebra is the
(associative) algebra of noncommutative polynomials; he answered a ques-
tion of Magnus—who had himself arrived at the solution—on the lower
central series of the free group. Some years earlier, in 1933, P. Hall had begun
commutator calculus in the free group, which led M. Hall to construct his
basis of the free Lie algebra; the link between the latter and the free group
is given by the work of Witt and the Magnus transformation.
In 1942 and 1944, Thrall and Brandt studied the free Lie algebra from the
point of view of representation theory of the linear group, and Brandt
computed the character—a formula closely related to the Witt formula.
At the end of the forties, Dynkin, Specht, and Wever simultaneously
discovered the characterization of Lie polynomials through the ‘left to right
bracketing mapping’. Some years later, Friedrichs gave his characterization
for Lie polynomials; his criterion fascinated many mathematicians, who
actually proved it.
After that, the subject was studied by many people, often independently.
Recently, it has had a new impulse, from the point of view of representation
theory of the symmetric group.
As far as we know, no book exists that exclusiyely treats free Lie algebras.
Bahturin, in his recent book on varieties of Lie algebras, devotes two chapters
to free Lie algebras; Bourbaki, Jacobson, and others, give an introduction
to the subject. It seems to us that the theory has become so extensive with
existing results so widely scattered, to justify the publication of a book on
the subject.
The book is partly written in the spirit of Lothaire’s Combinatorics on
words, with emphasis on the algebraic point of view; it can be considered as
a series of variations on Lyndon words; the presentation of the latter is rather
indirect, so the interested reader could begin by reading the corresponding
section by Lothaire.
In Chapter 0, we give without proof the Poincare Birkhoff Witt theorem,
viii
Preface
which enables us to prove that the Lie algebra of Lie polynomials is the free
Lie algebra; this necessitates a basis construction (in cases where the ring of
scalars is not a field), which is done through Lazard elimination.
The impatient reader may proceed directly to Chapter 1, where things
really begin. We give several characterizations of Lie polynomials, introduce
the shuffle product and present Hopf algebra-like properties of free associative
algebras.
Chapter 2 is devoted mainly to two results: subalgebras of free Lie algebras
are free; automorphisms of free Lie algebras are products of elementary
automorphisms. The related problem of characterizing free sets of Lie
polynomials is also treated.
In Chapter 3 we characterize exponentials of Lie series, and give several
results on the Hausdorff series, after having connected it to the canonical
projections of the free associative algebra.
The study of Hall bases begins in Chapter 4: we construct the Hall basis
of the free Lie algebra, and the corresponding Poincare-Birk hoff-Witt basis
of the free associative algebra. We show also that this basis construction is
identical to the one arising from Lazard elimination.
Chapter 5 gives some applications of Hall sets: the Lyndon basis, which
is a particular case of a Hall basis; the calculation of the dual basis in the
shuffle algebra; the construction of a Hall basis compatible with the derived
series of the free Lie algebra; and the order on the free monoid associated
with a Hall set and the associated codes.
In Chapter 6 we give some properties of the shuffle algebra: it is freely
generated by Lyndon words, and has a remarkable presentation. Related to
shuffles is the concept of subword. This leads to subword functions, a
generalization of binomial coefficients, and the Magnus transformation of
the free group. Commutator calculus is presented, and connected with the
Hall basis and the algebra of subword functions.
Chapter 7 studies circular words: after giving the formulas enumerating
them, we relate them to Hall sets. Two algorithms on Lyndon words are
presented, and we give a natural bijection between words on an ordered
alphabet and multisets of primitive necklaces.
The Lie representation of the symmetric group (or the linear group) is
considered in Chapter 8. Its character and the multiplicities of the irreducible
representations are given. Almost all of them occur. Remarkable Lie
elements, the Lie idempotents, are studied in the symmetric group algebra.
Representations on the components of the canonical decomposition of the
free associative algebra are also studied.
Chapter 9 shows the close connection between the free Lie algebra and the
Solomon descent algebra of the symmetric group. The primitive idempotents
of the latter represent the canonical projections, and the dimension of the
corresponding quasi-ideals has an interpretation in terms of necklaces. The
action on Lie monomials of elements of the descent algebra characterizes this
Preface	ix
algebra, which has as a natural homomorphic image the ring of symmetric
functions, and is itself dual to the ring of quasisymmetric functions.
Each chapter ends with an appendix: each subsection can be thought of
as an exercise, with hints or references, and gives some information on related
subjects; sometimes it is simply a review of related work.
Montreal	C.R.
July 1992
Acknowledgements
I first discovered the subject of this book in Gerard Lallement’s book on
semigroups and in Dominique Perrin’s chapter on factorization of free
monoids in Lothaire’s book; thanks to Adriano Garsia’s Combinatorics of
the free Lie algebra and the symmetric group, this subject was given a new
impulse, in the direction of algebraic combinatorics. While writing this book,
I had innumerable discussions with Marco Schutzenberger, who is for me
the initiator of the subject, and gave me useful advice as well as some
unpublished results. During this same time, Guy Melancon wrote his Ph.D.,
and was of considerable help, by discussions, reading and correcting the
successive versions of the manuscript. Paul Cohn kindly accepted this book
in the London Mathematical Society Monographs series and carefully read
the whole manuscript; he also communicated some unpublished results of
his Ph.D. thesis. I also thank, for many discussions and correspondence,
Sheila Sundaram, Andre Joyal, Pierre Leroux, Xavier Viennot, Hartmut
Laue, Pierre Bouchard, Francois Bergeron, and Nantel Bergeron. Special
thanks to Daniel Krob, who carefully read the manuscript, and found many
mistakes. I also want to thank the whole Department of Mathematics and
Computer Science of the Universite du Quebec a Montreal, for excellent
working conditions, and especially Dominique Chabot, Sonya Comtois,
Lucie Lortie, Marlaine Grenier, and Diane Amatuzio for their typing.
Finally, I was supported by a grant of NSERC (Canada) during the three
years I wrote this book.
Contents
Index of notation	xv
0. Introduction	1
0.1 The Poincare-Birkhoff-Witt theorem	1
0.2 Free Lie algebras	4
0.3 Elimination	7
0.4 Appendix	12
0.5 Notes	13
1.	Lie polynomials	14
1.1	Words, polynomials, and series	14
1.2	Lie polynomials	18
1.3	Characterizations of Lie polynomials	19
1.4	Shuffles	23
1.5	Duality concatenation/shuffle	26
1.6	Appendix	33
1.7	Notes	39
2.	Algebraic properties	40
2.1	The weak algorithm	40
2.2	Subalgebras	44
2.3	Automorphisms	45
2.4	Free sets of Lie polynomials	49
2.5	Appendix	50
2.6	Notes	51
3.	Logarithms and exponentials	52
3.1	Lie series and logarithm	52
3.2	The canonical projections	57
3.3	Coefficients of the Hausdorff series	61
3.4	Derivation and exponentiation	76
3.5	Appendix	80
3.6	Notes	82
xii	Contents
4. Hall bases	84
4.1 Hall trees and words	84
4.2 Hall and Poincare-Birkhoff-Witt bases	89
4.3 Hall sets and Lazard sets	98
4.4 Appendix	101
4.5 Notes	103
5. Applications of Hall sets	105
5.1 Lyndon words and basis	105
5.2 The dual basis	108
5.3 The derived series	112
5.4 Order properties of Hall sets	114
5.5 Synchronous codes	119
5.6 Appendix	124
5.7 Notes	126
6. Shuffle algebra and subwords	127
6.1 The free generating set of Lyndon words	127
6.2 Presentation of the shuffle algebra	129
6.3 Subword functions	131
6.4 The lower central series of the free group	136
6.5 Appendix	147
6.6 Notes	152
7. Circular words	154
7.1 The number of primitive necklaces	154
7.2 Hall words and primitive necklaces	158
7.3 Generation of Lyndon words	161
7.4 Factorization into Lyndon words	163
7.5 Words and multisets of primitive necklaces	166
7.6 Appendix	170
7.7 Notes	174
8. The action of the symmetric group	176
8.1 Action of the symmetric group and of the linear group	176
8.2 The character of the free Lie algebra	180
8.3 Irreducible components	185
8.4 Lie idempotents	194
8.5 Representations on the canonical decomposition	201
8.6 Appendix	206
8.7 Notes	215
Contents	xiii
9.	The Solomon descent	algebra	217
9.1	The descent algebra	217
9.2	Idempotents	224
9.3	Homomorphisms	233
9.4	Quasisymmetric functions	and enumeration of permutations	242
9.5	Appendix	248
9.6	Notes	254
References	256
Index	267

Notation
К
^К(Л), ^(Л)
М(Л)
t', t"
Id
ad(c)
(st”)
Un
A
Л*
1
A+ = Л*\1
IM
14
к<л>
к«л»
(S, P)
(S, w), (P, w)
(s, 1),(P, 1)
5
x
<5 = (id ® a) c <5
Ad
r
LU
6P
a~lS,a~lP
cone
<5'
sh
E
*
,<y
commutative ring with unit
free Lie algebra over К on the set A 4, 18
free magma on the set A 4
immediate left (right) subtree of the tree t 5
degree of the tree t 5
mapping у [c, y] 7
the tree (.. .((s, t), t),... ,t) 10
the submodule of an enveloping algebra generated by nth powers
of Lie elements 13, 57
alphabet, set of noncommuting variables 15
free monoid on Л 15
empty word, neutral element in Л* 15
15
length of the word w in Л*1.1; weight of the word w in T* 224
number of occurences of the letter a in w 14
K-algebra of noncommutative polynomials on Л, free associative
К-algebra on A 15
K-algebra of noncommutative formal series on Л 17
pairing К<<Л>> x К<Л> -> К 17
coefficient of the word w in S, P 15, 17
constant-term of S, P 17
coproduct a t—> a ® 1 4- 1 ® a 19, 52
principal anti-automorphism of К<Л> 19, 52
19, 52
K-algebra homomorphism К<Л> -> EndK(K<^>), ai—>ad(a) 19, 53
right normed bracketing, bracketing from right to left 20, 52
shuffle product 24
p-fold coproduct 25
26, 46
concatenation product K(A) ® K(A) K(A) 27
dual coproduct 27
shuffle product К<Л> ® К<Л>-> К<Л> 26
maps a polynomial to its constant term 27
convolution product in End(K(/l>) 28
complete tensor product К(Л)®К(Л) (shuffle at the left, con-
catenation at the right) 28
dual convolution product
xvi
Notation
concp
6'p
shp
I
Pu~l
S„
7Ti
D(w)
Uff
D£S
e
f(t)
н
h’h"
i\
Xi(s), pi(s)
T(s)
S„
C)
F(A)
M
I
F„(A)
lhn
nh
0(4")
Pn, Рл
st(w)
K<4>„
En
ch
In
л(Т)
D(T)
maj(T)
Xм
Хл
"-Л
Ui
/(>)
g°f
p-fold concatenation product K<4>®p -» K<4> 31
p-fold dual coproduct 31
p-fold shuffle product K<4>®p -> K<4> 31
left normed bracketing, bracketing from left to right 37
41
symmetrized product 57
symmetric group of order n
canonical projection K<4> -» U„ 58
canonical projection K<4> -> £f(A) 58
descent set of w 62
right action of permutation о on word w 63
sum of permutations whose descent set is contained in S 65
anti-automorphism <rt—><r~1 in KSn 65
foliage of tree t 84
Hall set 84
rewriting rule on standard sequences 86
standard factorization of Hall word h 85
Hall polynomial 90
mappings defined on standard sequences 91
derivation tree of a standard sequence 5 91
dual basis of K<4> 108
derived ideal of the free Lie algebra 112
binomial coefficient on words 131
free group on A
Magnus transformation F(A) -» Z<<4>>, at—> 1 4- a 132
infiltration product 134
lower central series of F(A) 136
Hall words of length <n 136
Hall exponent 136
series without term of degree < n 138
power sum symmetric function 156, 178
standard permutation of word w 167
space of homogeneous polynomials of degree n
multilinear part of K<4)„ 176
characteristic of a representation 178
characteristic of the nth Lie representation 180
shape of a standard tableau T 185
descent set of standard tableau T 185
major index of standard tableau T 185
irreducible character of Sn 186
value of irreducible character of S„ 188
space of homogeneous Lie polynomials of degree n 194
sum of permutations whose descent set is S 195
Dynkin-Specht-Wever Lie idempotent 195
Lie idempotent of Klyachko 196
subspace of K<4> 201
length of partition Л 201
plethysm of symmetric functions
Notation
xvii
hn qn Qc	complete homogeneous symmetric function 202 projection of К<Л> on its graded component of degree n 217 convolution product of the qn corresponding to the composition C 218
Г Г„ C(a) S(C) Ac К Я д T C(vr) /(w) M„ K, z(C), z(P) •i-M *P| к > p. с С	convolution subalgebra generated by the qn 217 graded component of Г 217 descent composition of permutation о 222 subset associated to a composition 223 is Dss(c> 223 Solomon descent algebra of S„ 223 is 0(Zn) 224 projection onto U, 224 is {г,, t2, t$,...} 224 composition associated with w in T* 224 length of w in T* 224 element of Q<r> 224 element of Q<r> 224 algebra homomorphism	-* Г, t, qt 225 partition associated to composition C, polynomial P 227 subspace of Q<T> 227 weight of polynomial P in Q<7"> 227 partition Л is finer than p 227 automorphism of Q<7’> 228 is C ° 7 228

о
Introduction
The aim of this preliminary chapter is to show that the Lie algebra of Lie
polynomials, which is introduced in Chapter 1 and which is the subject of
this book, is indeed the free Lie algebra. Apart from some elementary
universal constructions, two results are required: the Poincare-Birkhoff-Witt
theorem (actually only its consequence, that a Lie algebra is embedded in its
enveloping algebra), and the fact that the free Lie algebra is a free module.
The first result will not be proved here, as it is included in many textbooks.
The second one is proved in Section 0.3.
0.1 THE POINCARE-BIRKHOFF WITT THEOREM
Let К be a commutative ring with unit. A Lie algebra over К is a К-module
together with a K-bilinear mapping
x -> !£, (x, y) [x, y],
called a Lie product or Lie bracket, satisfying the two following relations, for
any x, y, z in Lf:
[x, x] = 0,	(0.1.1)
[[x, y], z] + [[y, z], x] + [[z, x], y] = 0.	(0.1.2)
The latter identity is called the Jacobi identity. Note that (0.1.1) implies
antisymmetry, i.e.
[x,y] + [y,x] = 0,	(0.1.3)
because 0 = [x + y, x + y] = [x, x] + [x, y] + [y, x] + [y, y] = [x, y] +
[y, x], by (0.1.1), bilinearity, and (0.1.1) again. In view of (0.1.3), we may
rewrite (0.1.2) as
[[x, y], z] = [x, [ y, z]] + [[x, z], у].	(0.1.4)
Subalgebras of Lie algebras and homomorphisms between Lie algebras are
defined as usual. Given an associative algebra over K, it acquires a
2
0 Introduction
Fig. 0.1
natural structure of Lie algebra when [x, y] is defined by
[x, y] = xy — yx.
Indeed, [x, y] is bilinear, (0.1.1) is immediate and (0.1.2) is easily verified.
Let У be a Lie algebra, and consider a Lie algebra homomorphism from
/.T into an associative algebra .ч/, with its natural Lie algebra structure.
Among all these algebras j/, there is one which has a universal property,
stated in the next result.
Proposition 0.1 Let <£ be a Lie algebra over K. There exists an associative
algebra si0 over К and a Lie algebra homomorphism q>0: -+ si 0 having the
following property: for any associative algebra si and any Lie algebra
homomorphism <p:	-> si, there is a unique algebra homomorphism f: si0 -► si
making the diagram of Fig. 0.1 commutative. The algebra siQ is unique up to
isomorphism.
This algebra .я/0 is called the enveloping algebra of J?.
Proof We first prove uniqueness up to isomorphism. Let six be another
couple like <p0, ,У0. Then, using Fig. 0.2, we deduce the existence of algebra
homomorphisms ft si0 -> six and gt six -+ .я/0 such that f ° (p0 = <Pi and
g°<Pi = <Po- Then id ° <p0 = <p0 = g ° (px = g°f acp0, and id, g°f are both
Fig. 0.2
0.1 Poincare-Birkhoff-Witt theorem	3
Fig. 0.3
algebra homomorphisms. By looking at Fig. 0.3, and by uniqueness, we find
that g ° f = id. Similarly, f ° g = id. This shows that ,c/0 and are
isomorphic.
We now prove the existence of за/0. Let T = T(ST} be the tensor algebra
of ST over K, that is
T(^) = © ^®".
n>0
Then T has a natural structure of associative algebra with unit. Let I be the
ideal of T generated by the elements x®y — y®x — [x, j](x, у e ST);
finally, let з/0 = T/I and <p0: ST -+ з/0 be the composition <p0 = p ° i, where
i is the canonical injection ST -+	and P the canonical surjective
algebra homomorphism T -+ T/I. Note that since Ker p = I and by the
definition of I, we have p([x, yj) = p(x ®y — y®x). Hence, for any x, у
in ST, (p0 ([x, Я) = p([x, у]) = p(x ® у - у ® x) = p(x)p( j) - pO)p(x) =
[p(x),	= [p° i(x), p° i(y)] = [<p0(*)> <Po(j)l Hence, <p0 is a Lie algebra
homomorphism.
Next, let з/ and <p be as defined in Proposition 0.1; then, since T is the
tensor algebra of ST, there is a unique extension of cp to an algebra
homomorphism (p:T->sf. Now, for, x, у e ST, we have <p(x ® у — у ® x —
[x, XD = <p(x)<p(У) ~ <p(y)<p(x) ~ 4>{\x, у]) = 0, because (p is a Lie algebra
homomorphism. Thus, I <= Ker ф, which implies that ф defines an algebra
homomorphism f\ sT0 = T/I -> st, defined by ф = f ° p. Now, for any x in
ST, we have f ° <p0(x) = f °p° i(x) = ф ° i(x) = <p(x), which shows that the
diagram in Fig. 0.1 is commutative. The homomorphism f is unique, because
(Po{ST) generates з/0.	□
The next result is the Poincare-Birkhoff-Witt theorem. We do not give a
proof, because many books contain one (e.g. Cartier 1954/55; Jacobson 1962;
Humphreys 1972; Dixmier 1974; Hochschild 1981; Lothaire 1983).
Theorem 0.2 Let ST be a Lie algebra over K, and suppose ST is a free
К-module with a totally ordered basis (Xi)ieI. Let be its enveloping algebra
4
0 Introduction
Fig. 0.4
and (pQ:	be the natural Lie algebra homomorphism. Then is a
free К-module with basis the set of decreasing products <р0(хь) • • • <Po(xin)(n 0,
ii,..., in g I, f > • • • > i„).
Corollary 0.3 Let <£ be a Lie algebra over K, and suppose it is a free
К-module. Let be its enveloping algebra and <p0: <£ -> .c/0 be the canonical
Lie homomorphism. Then <p0 is injective.
This result allows us to consider a Lie algebra as a Lie subalgebra of its
enveloping algebra, especially in the case when К is a field.
0.2 FREE LIE ALGEBRAS
Let JTq be a Lie algebra over K, A a set and i: A -+ a mapping. The Lie
algebra .У20 is called free on A if for any Lie algebra and any mapping
ft A -+ <£, there is a unique Lie algebra homomorphism f\ У20 -+ <£ such
that the diagram in Fig. 0.4 is commutative.
Theorem 0.4 For each set A, there exists a free Lie algebra ^f(A) on A,
which is unique up to isomorphism. Moreover, <S?(A) is naturally a graded
K-module, i is injective, the component of degree 1 of Jzf(A) is the free submodule
generated by A = i(A), and ^(A) is generated, as a Lie algebra, by A.
We denote the free Lie algebra by LK(A) or JT(A), and we also say that
=^(A) is freely generated by A. Recall that a magma is a set with a binary
operation. The free magma M(A) over A may be identified with the set of
binary, complete, planar, rooted trees with leaves labelled in A. Equivalently,
trees may be identified with well-formed expressions over A, which are
recursively defined by the following: each element of A is a well-formed
expression; if t', t" are well-formed expressions, then t = (t', t") is a well-
0.2 Free Lie algebras
Fig. 0.5
formed expression, which is identified with the tree obtained by taking a new
root, with immediate left subtree t' and immediate right subtree t". The binary
operation of M(A) is the mapping M(A) x М(Л) -+ М(Л), (f, t") i—► t. We
define the degree |t| of a tree t to be the number of its leaves, i.e. |t| = 1 if
te A and |(t', t")l = |t'| + |t"|.
Proof of Theorem 0.4 (i) We prove first uniqueness of the free Lie algebra.
Let i:A -+ and j: A -+ where У20, are free on A. By the diagrams
in Fig. 0.5 we deduce the existence of Lie algebra homomorphism j-.
and i: -+ such that j°i = j and i°j= i. By Fig. 0.6 we deduce that
id: =% -+ is the unique Lie algebra homomorphism such that id ° i = i.
Since we have i = i°j = i°j° i, we therefore must have i°j = id. Similarly,
j i = id, which shows that i, j are isomorphisms. Hence and are
isomorphic.
(ii) Let D(A) be the free (noncommutative, nonassociative) K-algebra over
A. One may view D(A) as the К-module freely generated by M(A), the free
magma over A. Multiplication in M(A) is linearly extended to D(A). An ideal
in D(A) is a submodule of D(A) which is closed under multiplication on the
left or right by any element of D(A). Let I be the ideal of D(A) generated by
the elements
(xy)z + (yz)x + (zx)y,	(0.2.1)
Fig. 0.6
6
0 Introduction
and
xx,
(0.2.2)
with x, у, z g D(A). Let &(A) be the quotient module D(A)II. It is immediate
that ^(A) has a multiplication inherited from D(Aj, and that with this
multiplication, &(A) is a Lie algebra over K. Moreover, with the canonical
mapping A -+ &(A), £f(A) is clearly the free Lie algebra on A.
Now, with the degree defined on M(A), D(A) is a graded К-module, with
О(Л)1 = ©aeAKa. Since the relations (0.2.1) and (0.2.2) are homogeneous of
degree >2 and since multiplication increases the degree, £f(A) is also a
graded module with ^(A\ = ©aeAKa.	□
Let л/0 be an associative algebra over К and j: A -> a mapping. Then
X) is called free on A if for any associative algebra and any mapping
/: A -► «я/, there exists a unique homomorphism of algebras f such that the
diagram in Fig. 0.7 is commutative. Uniqueness up to isomorphism is proved
as in the case of free Lie algebras, or enveloping algebras. Existence is shown
below, and a direct construction is done in Chapter 1. Let JTfA) be the free
Lie algebra on A, i: A -> JT(A) the corresponding mapping, and л/0 the
enveloping algebra of £f(A) with <p0: f?(A) ->• л/0 the corresponding Lie
algebra homomorphism. Then we have a mapping
j = cp0 A:A^ j/0.
Theorem 0.5 The enveloping algebra of the free Lie algebra £^(A) is a
free associative algebra on A. The Lie algebra homomorphism q>0: ^(A) ->
is injective, and cp0(JT(A)) is the Lie subalgebra of .g/0 generated by j(A).
Proof (i) Let л/ and f be as shown in Fig. 0.7. Then we have Fig. 0.8, in
which we show the existence of g and f, homomorphisms of Lie algebras
and associative algebras, respectively. By the universal property of the free
Lie algebra <T(A), we deduce existence and uniqueness of g. Then by the
universal property of the enveloping algebra, we deduce the existence of
0.3 Elimination
7
Fig. 0.8
/.To prove uniqueness of f, suppose we have the commutative diagram in
Fig. 0.7. Then, as above, we find a unique Lie algebra homomorphism g
such that g°i = f. Since j = <p0°i, we have f = f°j = f°(Po°i, and by
uniqueness of g, we deduce g = / ° <p0. Now, by the universal property of
the enveloping algebra, we deduce uniqueness of f.
(ii) In view of Corollary 0.3, it is enough to show that ^T(A) is a free
К-module to deduce that <p0 is injective. This will be done independently in
the next section (see Corollary 0.10). Now, the proof of Theorem 0.4 shows
that ^T(A) is generated, as a Lie algebra, by i(A). Hence, ср0(^(А)) is
generated by <p0° i(4) = j(A).	□
0.3 ELIMINATION
We begin by stating a theorem which allows us to ‘eliminate’ one variable.
Recall that a derivation of a Lie algebra is a linear endomorphism D such
that D([x, yj) = [Dx, y] + [x, Dy],
If c is an element of a Lie algebra JT, we denote by ad(c) the linear mapping
<£ -+ defined by ad(c)(y) = [с, у]. By (0.1.4) and (0.1.3), ad(c) is a
derivation of
Recall that A is canonically embedded in the free Lie algebra (4).
Theorem 0.6 Let c g A. Then the К-module ^f(A) is the direct sum of Kc
and of a Lie subalgebra which is freely generated, as a Lie algebra, by the
elements
(-ad(c))n(b), n>0, be A\c.
(0.3.1)
8
0 Introduction
Note that an element (0.3.1) is of the form [.. .[[b, c], c],..., c], with n cs.
The reader may verify that the subalgebra of the theorem is the Lie ideal
generated by Л\с.
We begin with a lemma.
Lemma 0.7 Let be a Lie algebra.
(i)	The set Der(Jzf) of derivations of is a Lie subalgebra of the algebra
of linear endomorphisms of .
(ii)	If V is another Lie algebra and h: -+ Der(Jzf) is a Lie algebra
homomorphism, then there is a unique Lie algebra structure on = <£ © <£',
extending that of and	and such that
Vx g	Vx' g	[x', x] = /i(x')(x).	(0.3.2)
(iii)	If is free on T, then each mapping T -+ extends uniquely to a
derivation of УТ.
The Lie algebra is called the semidirect product of and ST' with
respect to the homomorphism h: ST' -+ Der(^f).
Proof (i) Let D15 D2 be two derivations. We show that DX°D2 — If °D2 is
again a derivation. We have
D, - D/[x,)']) = D,([D,x, v] + [x,
= [D,D,x, y] + [D;X, D,y] + [Цх, D,y] + [x, ЦВД.
This implies that
[Di, ВД([х, y]) = D, о D2([x, y]) - D2 о D,([x, y])
=	у] + lx [£>i, аду)],
hence [D15 D2] is a derivation.
(ii)	It is clear that the Lie algebra structure on is completely defined
by (0.3.2), because the Lie bracket must be distributive and satisfy
[u, v] = — [и, и]. Conversely, define the bracket by (0.3.2). We verify the
Jacobi identity. We must show that
[[x + x', у + у'], z + Z]
+ [[у + y',z + z'],x + x'] + [[2 + z',x + x'],y + y'] = 0,
where x, y, z g ST, x', y', z' g ST'. By multilinearity and antisymmetry, we only
have to consider four cases:
(1)	[[*',/],/] + • • •
(2)	[[x', y’], z] + • • •
(3)	[[x', y], z] + • • •
(4)	[[x, y], z] + • • •
0.3 Elimination	9
Cases (1) and (4) are immediate consequences of the Jacobi identity in
and У. In case (2), we have, by antisymmetry:
[[%', y'], z] + [[y', z], x'] + [[z, x'], у'] = 6([x', y'])(z) - /i(x')((6(y')(z))
which is 0, because /i([x', y’]) = 6(x')6(y') — h(y')h(x'). In case (3), we have
[[< У], z] + [[y, z], x'] + [[z, x'l y] = [6(x')(y), z} - h(x')([y, z])
- [6(x')(z),y],
which is 0, because /i(x') is a derivation and by antisymmetry.
(iii) We use the previous part with the following: jSf = as a
К-module, with trivial Lie bracket, i.e. [x, y] = 0 for any x, у in
jSf' = Jf(T) as Lie algebra; 6(x')(x) = [x', x] for any x in <£' and any x in
where the Lie bracket is taken in the free Lie algebra ^(T); then h(x) is
a derivation of <£ (actually any endomorphism of is a derivation of
because the Lie bracket is trivial), and x'i-> /i(x') is a Lie algebra homo-
morphism (see Section 0.4.1). So^ = x gets a Lie algebra structure
with [(x, x'), (y, y')J = ([x', >0 + [x, У'Ъ [x', У])» where brackets are taken
in У(Т). Define a Lie homomorphism f : ^(T) -* by f(t) = (d(t), t),
where d is the given mapping T -+ exists, because У is free on T. We
may write /(x) = (D(x), u(x)) for any x in <£(T\ By the definition of the
product in u(x) is a Lie homomorphism ^f(T) -+ <£". Moreover,
u(t) = t for t in T. Hence, и is the identity and f (x) = (D(x), x). Then
№,y]), [x,y]) = /([x,y]) = [/(X),/(y)] = [(Dx,x), (Dy,y)] = ([Dx,y] +
[x, Dy], [x, y]), which shows that D is a derivation of Jz?(T) extending d. □
Proof of Theorem 0.6 Let = ^f(c), the free Lie algebra on c (which is
of dimension 1), and У = ^f(T), the free Lie algebra on T= N x B, with
В = A\c. By Lemma 0.7(iii), there is a unique derivation I) o! !/’ such
that D(n, b) = — (n + 1,6), for (n, 6) in N x B. The linear mapping
h: <£' -+ Der(J$f), c i—> D is a Lie homomorphism. Hence, by Lemma 0.6(ii),
we may define the Lie algebra = <£ © <£', whose product extends that
of <£ and <£', and such that [c, (n, 6)] = h(c)((n, b)) = D((n, b)) = — (n + 1, 6).
Now, since 3\A) is free on A, there is a unique Lie algebra homomorphism
1Д: ^f(/l) -+ such that <Д(с) = c, and ф(Ь) = (0, b) for be B.
Similarly, there are unique Lie homomorphisms <p':	-+ £^(A) and
cp: -+ Jz?(/1), such that <p'(c) = c and <p((n, 6)) = (— ad(c))"(6).
Let E be the subset of У defined by E = {x g \(p ° D(x) = [c, <p(x)]}. It
is a linear subspace, containing T: indeed, (p°D(n,b) = <p( —(n + 1,6)) =
-(-ad(c))n +1(b)= —[( — adc)n(b), c] = [c, <p(n, 6)]; moreover, since
10	0 Introduction
у|—► [с, у] is a derivation, Е is a Lie subalgebra of У'. х, у е Е implies
д> ° D([x, у]) = <p([Dx, у] + [х, Dy]) = [<р ° Dx, <ру] + [<рх, <р ° Dy]
= [fc, <рх], <ру] + [<рх, [с, <ру]] = [с, [<рх, <ру]]
= [с, <р([х, у])].
Непсе, Е is equal to <£, and we have <p ° Dx = [c, <px] for any element of
Define a linear mapping a: -+ ^(Л) by a(c) = c and a(x) = <p(x) if x
is in . Then a is a Lie homomorphism: indeed, we have a([c, x]) =
a(/i(c)(x)) = a(Dx) = <p ° Dx = [c, <px] = [a(c), a(x)].
The homomorphisms i/r°a and a°i/r are the identity of and ^(Л),
respectively. Indeed, ф ° a(c) = <Д(с) = c, and ф ° a((n, b)) = ф ° <p((n, b)) =
<Д(( — ud(c))"(b)) = — (аЛ(фс))п(фЬ) (because ф is a Lie homomorphism) =
(— ad(c))"(0, b) = (n, b) (by a straightforward induction, using the definition
of the Lie bracket in J^); since c and the (n, b) generate the Lie algebra JS^,
we deduce that ф ° a is the identity of . A similar argument shows that
a ° ф is the identity of &(A).
Thus, a is a Lie isomorphism -+ &(A).
Observe that a maps T onto the set of elements (0.3.1), and onto Kc.
Since = <£ © JS?' as a К-module and since JZ is free on T, the theorem
is proved.	□
Consider again the free magma M(A) (see Section 2). There is a canonical
mapping ф: M(A) -> ^(Л), which is the identity on A and which sends each
tree (t', t") to [фЬ, <Дг"]. If s, t are two trees, we denote by (stp) the tree
(.. .((s, t), t),..., t). Consider trees t0,..., tn and subsets T0,...,T„+l of
М(Л) with
to e = >
G e Л = {(ttg) | p > 0, t g T0\t0},
tneTn = {(tt^JIp^O^Gr^At^J,
Tn + l = {(ttpn)\p>Q,tcT„\t„}.
(0.3.3)
Corollary 0.8 With the above notations, one has a direct sum decomposition
(as K-module)
^(Л) = Кф(10) ©© Кф(1„) ©&n + 1,
where ^n + l is a Lie subalgebra freely generated by ф(Т„ + 1).
Proof For n — — 1, this is clear. Let us assume that this is true for n — 1
(n > 0), and we prove it for n. Since <£n is freely generated by ф(Тп), we have
0.3 Elimination	11
Now, we have t„ g T„ and Tn+l = {(tt^)\p > 0, t g Tn\t„}. Thus,
by Theorem 0.6, ^(Tn) is the direct sum Kt„©^', where jjf' is freely
generated by the elements [.. .[[t, tj, t J,..., t„] in ^(T„), t g T„\t„. Return-
ing to ^(Л), we find that <£n = Ki//(tn) © JS£+ n where &n + l is freely
generated by ф(Тп + 1).	□
We say that a subset E of M(A) is closed if for each tree (t', t") in E, one
has t', t" g E. In other words, E contains all the subtrees of its elements. A
Lazard set is a totally ordered subset L of Л/(Л) such that for any finite,
nonempty and closed subset E of М(Л), one has
L n E = {t0 > tj > • • • > t„},	(0.3.4)
for some n > 0, such that (0.3.3) holds, and that moreover
Tn+1r\E = 0.	(0.3.5)
Corollary 0.9 IfL is a Lazard set, then ф(Ь) is a basis of the К-module (A).
Proof The set ф(Ь) is linearly independent: indeed, it is enough to show
that it is the case for each finite subset К of E; then we can find a finite
closed subset E 0 containing K, and it is enough to show that ф(Е n E)
is linearly independent; this is a consequence of (0.3.4), (0.3.5), and Corollary
0.8.
Now, let P 0 be in ^f(4); the latter is linearly generated by ф(М(А)),
hence we may find a finite nonempty subset В of A such that P is a linear
combination of <Д(М(В)). Denote by d the total degree in the variables in B.
Let d(P) = d, and define
E= {t g M(B)\d(t) < d}.
Then E is a nonempty, closed and finite subset of М(Л), so we have (0.3.4),
(0.3.3), and (0.3.5). Note that for each t = t0,..., t„, d(t) < d, and, hence, also
d(<A(O) < d because ф does not increase degrees. We have, by (0.3.5),
t g Tn + i => d(t) > d. Hence, since ф is homogeneous, each nonzero element
of ф(Т„+1) is homogeneous of total В-degree >d. Since the Lie bracket is
homogeneous, the same holds for each homogeneous component of each
element of the subalgebra generated by ф(Тп+1). Hence, by Corollary
0.8 and the degree assumption on P, the latter is a linear combination of
<А(г0),...,1Д(гя).	□
Corollary 0.10 ^(A) is a free K-module.
Proof In chapter 4, we shall define Hall sets, show that they exist
(Proposition 4.1) and that each Hall set is a Lazard set (Theorem 4.18(i)):
12
0 Introduction
these two results will be proved independently of this chapter. Hence, Lazard
sebexist, which implies by Corollary 0.9 that У(А) is a free K-module. □
0.4 APPENDIX
0.4.1 Variants of the Jacobi identity
The Jacobi identity (0.1.2) may be rewritten, using (0.1.3):
[X,	= [[x, y], z] + [y, [x,z]].
This means that the linear endomorphism ad(x): у i—> [x, y] is a derivation
of the Lie algebra У2, i.e.
ud(x)([y, z]) = (ad(x)(y), z~] + [y, ad(x)(z)].
The Jacobi identity may also be written
[*, [У,2]] - [у, [X, z]] = [[x,y],z].
This is equivalent to
(ad(x), ad(y)](z) = ad([x, y])(z),
and means that x i—> ad(x) is a Lie algebra homomorphism -► End(jSf).
Another equivalent form of the Jacobi identity is
[x, [y, z]] = [[x, y], z] - [[x, z], у].
This may be used to show by induction that if a set X generates a Lie algebra,
then the latter is linearly generated by the left to right bracketed elements
(or left normed elements)
[.. .[[x15 x2], x3],..., x„], Xi g X.
0.4.2 Witt formula
Let A have q elements and a„ denote the dimension of the homogeneous
component of degree n of ^(A) over a field K. Then Theorems 0.2 and 0.5
imply that
п	= E p.x",
n > 1	л > 0
where /?„ is the dimension of the component of degree n of the free associative
algebra on A. Now, it is not difficult to see that /3„ = qn (see Section 1.1).
Then, by taking the logarithm of the previous formula and by Mobius
0.5 Notes	13
inversion, one obtains the formula of Witt (1937):
= - E /'(W7*'-
П d | л
0.4.3 Canonical decomposition
Let <£ be a Lie algebra over K, and its enveloping algebra. We assume
that J2? is a free К-module, hence <£ is embedded in л/. Suppose that К
contains Q and define Un as the linear span of the elements
Г	ч 1 V
(x1,...,x„) = - X *,(1) • • • *<7(л)>
n! <reS„
for each choice of xn ..., xn in <£ (S„ is the symmetric group). Then Uo = K,
Ux = & and
= © U„.	(0.4.1)
л>0
Indeed, denote by Vn the linear span in л/ of the elements xv .. ,xp, with
p < n and x, g Jzf. Then
x1...xn = (x1,...,x„)mod Vn.x.	(0.4.2)
This is a consequence of the formula xy = [x, y] + yx, which implies that
all products xff(1)... xa(n} are congruent to Xj ... xn mod К-p Summing
up, we get (0.4.2). Now, take the хг in an ordered basis В of JZ. Then
eqn (0.4.2) gives triangular relations between the elements x1...xn
(хг g B, Xj > • • • > x„) and the elements (xn ..., x„). Hence, Theorem 0.2
implies that the latter form a basis of <s/, which implies (0.4.1). Note that
Proposition 3.6 shows that Un coincides with the submodule of generated
by the elements x", x g .
0.5 NOTES
In Sections 0.1 and 0.2, we have followed Bourbaki (1972). Note that
Corollary 0.3 is true under weaker hypotheses (Cohn 1963). For the Lazard
elimination process (Section 0.3), we have followed Viennot (1978). Theorem
0.6 and its corollaries are due to Lazard (1960).
1
Lie polynomials
After introducing words, noncommutative polynomials and series, we define
Lie polynomials. One of the main results presents the various characteriza-
tions of Lie polynomials. This naturally leads us to define the shuffle product,
and to study the duality between concatenation and shuffle product; in other
words: the Hopf-algebra-like properties of the free associative algebra.
Related questions are treated in the appendix, including the support of the
free Lie algebra (the set of words which may appear in Lie polynomials),
the free Lie p-algebra and the Jacobson identities, the kernel of the left to
right bracketing mapping and a brief excursion into automata theory.
1.1 WORDS, POLYNOMIALS, AND SERIES
Let A be a set, which we call an alphabet, whose elements are letters', for the
main applications, the alphabet will be finite, but it is convenient to admit
infinite alphabets. A word on the alphabet A is a finite sequence of elements
of A, including the empty sequence, called the empty word. With the
concatenation product, the set of all words over A gives rise to a monoid
called the free monoid on A and denoted by A*; the neutral element is the
empty word, denoted by 1. The set of nonempty words is denoted by A + .
Each letter is itself a word, and each word w is the product of its letters,
from left to right:
w = ava2   (ate A).
Here, n is the length of w, denoted by |w|. For each letter a, we denote by
|w|a the number of occurrences of a in the word w: it is the length of w with
respect to the letter a.
A factor of a word w is a word и such that w = xuy for some words x, y,
if, moreover, x = 1, и is called a left factor, or a prefix of w; a right factor
(or suffix) is defined similarly. A factor и of w is called proper if и / w, and
nontrivial if и / 1.
The terminology ‘free monoid’ is justified by the following universal
property.
1.1 Words, polynomials, and series
15
Proposition 1.1 For any mapping f from A into a monoid M, there is a unique
extension of f to a monoid homomorphism f : A* -+ M such that the diagram
in Fig. 1.1 is commutative (where i is the natural injection A -+ 4*).
Proof If w = ara2 .   a„(at e A) then clearly we must have f(w) =
f(ai)f(a2) - - • Л“Л So f is unique. Moreover, it is easily verified that if f
is defined in this way, then 7 is a monoid homomorphism and satisfies
f = f°i.	□
Let К be a commutative ring with unit. A noncommutative polynomial on
A over К is a linear combination over К of words on A. We simply say
polynomial when no confusion arises. If P is a polynomial, we write it as
P= £ (P, w)w.
we A*
Thus, (P, w) is the coefficient in P of the word w; all but a finite number of
the (P, w)'s are zero. The set of all polynomials is denoted by K<4). It has
a K-algebra structure, with componentwise addition, and product defined by
(PQ,y»)= z (p.uxe.0.
w = uv
In other words, it is the K-algebra of the monoid A*.
As we shall consider other products on K<A>, we call the product just
defined the concatenation product and K<4> with this product the concatena-
tion algebra. Note that A* is contained in K<4>, as a submonoid: K<4> is
a free К-module with basis A*. The algebra K<4> is the free associative
K-algebra generated by A, as is shown by the following universal property.
Proposition 1.2 For each mapping f from A into an associative K-algebra
, there is a unique extension off to a K-algebra homomorphism f: K(Aj> -> .<з/
such that the diagram in Fig. 1.2 is commutative (where i is the natural injection
AK(Af).
16
1 Lie polynomials
Proof If f is such an extension, then f\A* is a monoid homomorphism
A* -> jrf (multiplicative structure of зз/), so f\A* is unique by Proposition
1.1. Since A* linearly generates over K, f must be unique.
Now, denote by g the extension of f to a monoid homomorphism
A* -> stf (g exists by Proposition 1.1). Define f to be the linear extension
of g to зз/: it exists because A* is a basis of the К-module Х<Л>. Given
two polynomials P = ^ueA, (P, u)u, Q =	(Q, v)v, we have PQ = ^u,v
(P, u)(Q, v)uv, hence
/те) = Е(Л«)(С, »)»(«>)
U, V
= E (p’ U)(Q’ v)g(u)g(v)
U, V
= E (p’ MW) E (G’ v^(v)
U	V
which shows that f is an algebra homomorphism. Evidently, fi = f □
The degree of a nonzero polynomial P is
deg(P) = sup{|w|, w g A*, (P, w) / 0}
and deg(O) = — oo. Similarly, for a letter a in A, the partial degree of P with
respect to a is
dega(P) = sup{|w|a, w g A*, (P, w) / 0}
and dega(0) = - oo.
A polynomial P is homogeneous of degree n if P is a linear combination
of words of length n, and finely homogeneous if P is a linear combination
of words having all the same partial degrees with respect to all letters;
homogeneous components and finely homogeneous components are defined
as usual.
1.1 Words, polynomials, and series	17
A formal series (or series) on A over К is an infinite formal linear
combination
S = £ (S, w)w.
we A*
The constant term of S is (S, 1). The set of all series is denoted by K((A>>.
It acquires a K-algebra structure, with product
(ST,w)= Y (S,u)(T,v).
w — uv
As before, we use the word ‘concatenation’ when confusion with other
products may arise. Note that К<Л> is a subalgebra of К<<Л>>.
There is a natural duality between К<Л> and К<<Л». Indeed, define the
pairing
К«Л» x К<Л> -> К,
(S,P)^(S,P)= E (S,w)(P,w).
we A*
This sum is finite because P is a polynomial. It is easily seen that with this
pairing,	may be identified with the dual space of К<Л>. When
restricted to K(A), this pairing gives a scalar product on К<Л> with A* as
orthonormal basis.
We need some topological remarks, but we shall not go into too much
detail. Put on К the discrete topology, and consider on K<<4>> the smallest
topology such that each mapping
Si->(S,w),	K«A»->K,
is continuous. Equivalently, a fundamental system of neighbourhoods of a
series S is the family of subsets, indexed by finite subsets L of A*,
VL(S) = {TeK((A)) | VwgL,(P, w) = (S, w)}.
Then	becomes a complete topological ring, and К<Л> is a dense
subring of	This topology is sometimes called the A-adic topology.
When A is finite, it is defined by the ultrametric distance
d(s, t) =
for some fixed 6,0 < в < 1, where for any nonzero series S, co(S) is the length
of the shortest word w such that (S, w) / 0, and w(0) = + co.
This topology has the following nice property, with respect to infinite sums:
if (5,)Ie/ is a family of series such that for each neighbourhood of 0, all but
a finite number of these series are in this neighbourhood, then the family
18	1.2 Lie polynomials
(S^ is summable, and its sum 5 is defined for any word w by
(S, w) = £ (S„ w).
ie I
Observe that only finitely many terms in the right-hand sum are nonzero,
by hypothesis. This condition may also be expressed by saying that the family
(5,) is locally finite. In particular, if a series S has constant term 0
(equivalently co(S) > 1), then each family (a„S")„>o is summable, and one
may define a„S". In particular, (1 — S)-1 = £„>0 Sn, and if К contains
Q, one defines
СЛ
exp(S) = es = £ —, log(l + S) = £ ------------Sn,
n>o n!	„>i n
and one has the usual formulas
log(es) = S, exp(log(l + S)) = 1 + S.
Similar consideration apply to infinite products, and to other rings of
formal series, like the complete tensor product
1.2 LIE POLYNOMIALS
Given two (noncommutative) polynomials P, Q in К<Л>, their Lie bracket
(or Lie product) is as usual defined by
[Л Q1 = PQ-QP-
A Lie polynomial is an element of the smallest submodule of К (Ay
containing A and closed under the Lie bracket. By Theorem 0.5 it is the free
Lie algebra on A, so we denote it by ^K(A) or 5£(A). Moreover, K(Ay is
the enveloping algebra of ^K(A).
The next result is elementary, but useful. If L is another commutative ring
with unit and <p: К -> L a Z-linear mapping, then we still denote by <p the
mapping К<Л> -+ £<Л> defined by <p(P) = £we?1* <p((P, w))w. Note that if
(p is a ring homomorphism, then so is its extension to K(Ay.
Lemma 1.3 (i) ^(A) is finely homogeneous, that is, if P is a Lie polynomial,
then each finely homogeneous component of P is a Lie polynomial.
(ii) With L and <p as above, <p(P) is a Lie polynomial in L(Ayfor each Lie
polynomial P in К (Ay.
Proof (i) is true when P is a letter. Moreover, if it is true for P, Q, and if
cte K, then it is also true for P + Q, ctP and [P, Q] = PQ — QP. So it is true
for any Lie polynomial.
1.3 Characterizations of Lie polynomials	19
(ii) If (p is a ring homomorphism, then so is its extension to K<A>, and
clearly the lemma holds in this case. Let и: Ж -> K,v: £ -> L be the canonical
ring homomorphisms sending 1 to 1. Then each Lie polynomial P e &k(A)
may be written as a finite sum
P = Y.<‘I“(PI),	(1.2.1)
where a,- g K, P{e This is clear when P is a letter, and is easily
extended by induction to all of &K(A).
Now, if QgZ<A>, then for a in K, (p(au(Q)) = cp(ct)v(Q); indeed Q =
^n7w7(n7eZ, w7g Л*), hence <p(au(0) = <p(^ n7avv7) = <p(n7a)vv7 =
£ n7<p(a)w7 = <p(a)u(0, <p being Z-linear. Thus, by (1.2.1) and linearity of <p,
<p(P) = X <P(aiu(-f/)) = E ф(а/М^)- Since Pt is in J^Z(A) and as v is a
ring-homomorphism, we conclude that и(Д) is in У\(А)', hence so is <p(P).
□
1.3 CHARACTERIZATIONS OF LIE POLYNOMIALS
We shall characterize Lie polynomials among the set of all polynomials. For
this, we need to define several linear mappings.
Define a homomorphism of K-algebras (sending 1 to 1)
<5: К<Л>-> К<Л>®кК<А>,	<5(a) = a ® 1 + 1 ® a,
for any letter a. Such a homomorphism exists by Proposition 1.2.
Define a linear mapping a from К<Л> into itself, in the following way:
for any word w = aY ... a„(a{g A), let a(w) = ( — l)"a„.. .av In other words,
a is the anti-automorphism of K<A) which sends each letter «to —a.
In particular, a(PQ) = a(Q)a(P) for all polynomials P, Q. Now, define
J = (id® a) ° d, where id is the identity of К<Л>. For any polynomial P,
recall that ad(P) denotes the linear mapping K<A) -+ К<Л) defined by
ad(P)(Q) = [P, Q] = PQ- QP.	(13.1)
Moreover, define a homomorphism of K-algebras Ad from К<Л> into
End(K</l» by
Ad(a) = ad(a)
for any letter a; Ad exists by Proposition 1.2. Note that Ad # ad in general:
e.g. ad(ab) is the endomorphism Q i—> [ah, Q] = abQ — Qab, while Ad(ab) is
the endomorphism Q i—> [a, [b, Q]] = abQ — aQb — bQa + Qba.
Let D: К (A) -> K(A> be the linear mapping which sends each word w
of length n into nw. It is easily verified that D is a derivation of KfAy, that is,
D(PQ) = D(P)Q + PD(Q).
20	1 Lie polynomials
Actually, D is the unique derivation of К (A) such that D(a) = a for any
letter a.
Finally, define the ‘Lie bracketing from right to left’ or the ‘right normed
bracketing’ to be the unique linear endomorphism r of К<Л> such that
for any word w = a{ ... an of positive length, one has r(w) = [a15...,
[u„_15 «„]. .Moreover, r(l) = 0. For example, one has for a, b, c in
A: r(abc) = [a, [b, c]] = [a, be — cb] = abc — acb — bca + cba.
In the next result, the ring К is assumed to be a Q-algebra, that is, a ring
containing Q as a subfield. We shall also assume that A has at least two
letters; the theorem is trivial in the one-letter case (except for condition (ii),
which is not equivalent to the others).
Theorem 1.4 For P in K(A), the following conditions are equivalent:
(i)	P is a Lie polynomial',
(ii)	ad(P) = Ad(P) and (P, 1) = 0;
(iii)	b(P) = P ® 1 + 1 ® P;
(iv)	b(P) = P ® 1 - 1 ® P;
(v)	(P, 1) = 0 and r(P) = D(P).
The equivalence of (i) and (iii) is often expressed in the following way: let
A' be a copy of A, with bijection a i—► a'; let each letter in A commute with
each letter in A'. Then, denoting a polynomial P by P(a, b,...), (iii) is
rewritten as P(a + a', b + b',...) = P(a, b,...) + P(d, b',...). Thus, a poly-
nomial is a Lie polynomial if it is additive, in the above sense.
Theorem 1.4 is still true if К is a Z-algebra without torsion (see Corollary
4.17). However, it is not true if К is a field with nonzero characteristic; when
the characteristic is a prime number p, it characterizes the free p-Lie algebra
(see Section 2.5.2).
Lemma 1.5 Define two linear mappings A, cone: K(A) ® KfA) -> KfA)
by z(P ® Q) = D(P)Q and cone (P ® Q) = PQ. Then for any polynomial P,
one has 2°<5(P) = r(P) and cone ° <5(P) = (P, 1).
Proof We have b(l) = 1 ® 1 and D(l) = 0 so that z°<5(l) = 0 and
cone ° <5(1) = 1. It remains to show that for n > 1 and an,..., аг in A, one has
Л°д(а„... аг) = r(an.. .af),	(1.3.2)
and
cone ° b(a„... aj = 0.	(1.3.3)
We do it by induction on n. The case n = 1 is easy:
b(«!) = (id (x)a)°b(«i) = (id ® a)(aj ®1 + l(g)zi1) = zz1<g>l — 1 ® aj,
1.3 Characterizations of Lie polynomials	21
so that
A ° 5(aJ = D(ar) — £>(1)^ = аг = г(аг)
and
cone ° 5(aJ = at — at = 0.
Suppose that eqns (1.3.2) and (1.3.3) are true for n > 1; we prove them
for n + 1. Let
. ,«i) =	Qt
i
Then S(an ...«i) =£-Д® a(Q,), hence we have by induction
E D(Pi)<*(Qi) = Ao d(a„ ...ar) = r(a„... aj,	(1.3.4)
and
E Pi°L(Qi) = cone ° d(a„... af) = 0.	(1.3.5)
Now, we have
<5(«n+i .. .«J = (id® a)o<5(an+1 .. .aj
= (id ® ct)(d(a„+1)d(a„... a J)
=	(id ® a)((an+ j ® 1 + 1 ® an+ j)(£ Pf ® Q,))
=	(id ® a)(£ «„ +1^- ® Qi + £ Л ® «л+ iQ.)
=	E a» + ipi ® a(Gf) “ E pi ® a(&K +1,
because a is an anti-endomorphism of the algebra К<Л> and a(an+1) =
— an+l. Thus, we have
... af) = £ D(an + 1^.)a(Qi) - £ WXQ.K+i
= E «n + i^(Q.) + E	- E WMQ.X + i
= 0 + a„+lr(an ...aj- r(a„ ...a1)a„ + 1
= [a„+1,r(a„.. .ui)]
= r(a„+1... aj,
where we have used in the second equality the fact that D is a derivation
such that D(an+ J = an+ n and (1.3.4) and (1.3.5) in the third one. Moreover,
we have
cone ° <5(u„ 4-i ...Ui) = £ «л +1 ^f^(Qi) E	+ i 0»
by (1.3.5).
□
22
1 Lie polynomials
Define a linear mapping p: K(A) ® K(A) -> End(K</l» by
pU\®P2)(Q) = PiQP2-
Lemma 1.6 (i) For any polynomial P, one has ad(P) = p(P (x) 1 — 1 (x) P)
and Ad(P) = ц°<5(Р).
(ii) p is injective if A has at least two letters.
Proof (i) The first equality is immediate. The second one holds if P is a
letter, and so it is enough to show that p ° d is multiplicative, because Ad is,
so both are algebra homomorphisms which coincide on the generators, and
thus are equal. Now, p ° 6 = p ° (id ® a) ° d, and d is multiplicative. A routine
verification shows that p ° (id ® a) is multiplicative:
p о (id ® a)((Pt ® Р2МГ ® Q2))(P) = p ° (id ® «XPiQi ® P2Q2)(P)
= р(Ле1®а(е2)а(^))(Ю
= P1Q1Pa(Q2)a(P2)
= M(P1®a(P2))(Q1Ra(Q2))
= p(J\ ® ^PJ^plQ^ ® a(Q2))(K)
= (^o(id®a)(P1®P2))
°(^°(id ® a))(Qi ® Q2)(K).	□
(ii) Note that A* x A* is a basis of the free К-module K(A) (x) К<Л>.
Take an element x # 0 in this space; it may be written x = <i<n * ut ® v(,
where * indicates a nonzero coefficient, and (uf, v{) are distinct couples of
words. Let ur be of minimal length among all the uh take N greater than
the lengths of all these words, and let a, b be two distinct letters in A. We
show that u1aNbv1 is different from each word for i > 2: this will
imply that p(x)(aNb) / 0, hence p is injective. Suppose u1aNbvl = ща'^Ьь^.
Then, by minimality of u15 ux is a left factor of uf: uf = urs => aNbvt = saNbvt.
Because N is big, s is a left factor of aN, hence s = aj, which implies
aNbvr = aj+NbVi. Since a / b, we must have j = 0; therefore = uY and
Vj = vt.	□
Lemma 1.7 a(P) = — P for any Lie polynomial P.
Proof This is clear for each letter, by definition of a. Furthermore, if this
equality is true for polynomials P, Q, and if a e K, then it is also true for
P + Q, aP and for [P, QJ: indeed, a([P, Q]) = a(PQ - QP) = a(Q)a(P) -
a(P)a(0 = ( —Q)( —P) — ( —P)( —Q) = — [P, Q], because a is an anti-
endomorphism of the K-algebra К<Л>.	□
1.4 Shuffles
23
Proof of Theorem 1.4 (i) => (ii) The set of polynomials P satisfying ad(P) =
Ad(P) is a submodule of К (A) containing A and closed under Lie bracket:
indeed if ad(P{) = Ad(Pi), i = 1,2, then for any polynomial Q
ad([Pv P2])(Q) = ad(P.P2 - P2P^Q)
= P1P2Q - p2p& - QP,p2 + Qp2p^
and
P2])(Q) = Ad(P,P2 - P2P^Q)
= (Ad(J\) > Ad(P2) - Ad(P2)^Adm)(Q)
= (ad(P1) о ad(P2) - ad(P2) о ^(PJXQ)
= LA, [P2, Q]] - [P2, ЕЛ, QUl
= Pffl2Q - PrQP2 - P2QPr + QP2Pr
- P2P,Q + P2QPr + PrQP2 - QPrP2
= p,p2Q + QP2P, - p2p,Q ~ QPffli-
Hence, ad([Pi, P2]) = Л^([Р15 P2]) and the set of polynomials satisfying (ii)
contains all Lie polynomials.
(ii)	=> (iv) We have by hypothesis and Lemma 1.6(i) that fflP ® 1 —
1 0 p) = ц о <5(P). Then by Lemma 1.6(ii), we obtain P ® 1 — 1 ® P = d(P).
(iv)	=> (v) Apply the mappings z and cone of Lemma 1.5 to the equality
J(P) = P ® 1 - 1 ® P. We obtain r(P) = D(P) and (P, 1) = P - P = 0.
(v)	=> (i) Write P as £„>0 P„, where P„ is the homogeneous component
of degree n of P; then r(P) = D(P) = £ nP„ (by definition of D). Thus
nP„ = r(P)„, the homogeneous component of degree n of r(P). Since r(P) is
evidently a Lie polynomial and since &(A) is homogeneous by Lemma 1.3(i).
nPn is a Lie polynomial. Finally (since we may divide by n in K), Pn is a Lie
polynomial, for each n > 1. To conclude, observe that Po = (P, 1) = 0.
(i) => (iii) We have shown that (i) => (iv). Hence, if P is a Lie polynomial,
we have <5(P) = (id ® a)°5(P) = (id ® a)(P® 1 — 1 ® P) = P ® 1 + 1®P,
because a is an involution, and by Lemma 1.7.
(iii) => (v) We have 3(P) = (id ® a) ° д(Р) = P ® 1 + 1 ® oc(P). Applying
the mappings z and cone of Lemma 1.5 to this equality we get r(P) = D(P)
and (P, 1) = P + a(P). Taking constant terms, and observing that (a(P), 1) =
(P, 1), we have (P, 1) = 2(P, 1), hence (P, 1) = 0.	□
1.4 SHUFFLES
Let w = aY ... a„ be a word of length n and I be a subset of {1,..., n}. We
denote by w\l the word ah ... aik, if I = {ix < i2 < • • • < ik}; in particular,
w\I is the empty word if / = 0. Such a word w\l is called a subword of w.
1 Lie polynomials
24
Note that when
= U Ij,
j=i
then w is determined by the knowledge of the p words w|/7 and the p subsets
Ij. Given p words ur,..., up of respective lengths n15..., np, their shuffle
product, denoted by u{ lli - • -lliup, is the polynomial
W1 LU - • -LUUp = X	Ip)
where the sum is extended to all p-tuples (/15..., Ip) of pairwise disjoint
subsets of {1,..., и} (n = nx + • • • + np) such that
and |/j| = n} for any j = 1,..., p, and where the word w = w(Ilf... ,Ip) is
defined by w\Ij = Uj for j = 1,..., p. Note that иг ш• • -шup is the sum of
words of length n, so it is a homogeneous polynomial of degree n. In
particular, if one of the Uj is empty, it may be omitted without changing the
shuffle product. Moreover, the product does not depend on the order of the
words Uj, as the reader may verify.
A word appearing in the shuffle product uY ш  • • ш up is called a shuffle of
«i,..., up. Thus, a shuffle of иг,..., up is a word obtained by ‘shuffling’
together the words ut,..., up without changing each word Uj. The shuffle
product is then the sum of all shuffles, with multiplicities. For example, with
a, b, c 6 A, we have: ab ш ас = abac + 2aabc + 2aacb + acab.
The shuffle product ш is extended to K<A> and K<<4>>, by the formula
51lu---lu5p = Y (5i> Mi) •  • (Sp, wp)wx ш • • • uj up.
Ml, ... ,Up
This infinite sum makes sense, because it is locally finite: indeed, for any
word w, there are only finitely many p-tuples (uY,... ,up) such that w is a
shuffle of u15..., up. If each Sj is a polynomial, then so is their shuffle product.
Note that ur ш (w2 ш w3) = (ux ш w2) ш w3 = иг ш u2 ш w3. Hence, the
shuffle product is associative. As it is also clearly distributive with respect to
addition, we obtain on K<A> and K<(4>> a commutative algebra structure
called the shuffle algebra. The neutral element is the empty word.
The shuffle product is intimately related to the homomorphism 6 of the
previous section. We consider the complete tensor product
fP=K(A>
1.4 Shuffles	25
with its concatenation structure. An element of #p is an infinite linear
combination
E aui....,UpWi ® • • -®«p-
ui, ..., upeA*
The product in Jp is defined by (iq ®	®	®	® vp) =
(«1^1) ®	® (upvp) and extended by infinite linearity (this always makes
sense).
Define an homomorphism of concatenation algebras dp. K((A>> ->
by bp(a} = a ® 1 ®	® 1 + 1 ® a ®	® 1 + ••• + 1 ® 1 ®	® a for
any letter a. Indeed, dp so defined extends to a unique algebra homomor-
phism К<Л> -> /р (by the universal property of K<4>, Proposition 1.2);
this extension being homogeneous and degree-preserving, it may be extended
by infinite linearity to К<<Л>>. Note that d = <52.
The following result shows that dp has another equivalent definition, in
terms of the shuffle product.
Proposition 1.8 For each series S, one has
dp(S) =	(S, иг ш-• -Lu up)u1 ® up. (1-4.1)
Ul ,..., upe A *
Recall that the scalar product of a series 5 and a polynomial P is
(S,P)= X (МЛ4
WE A*
Proof We have only to show that eqn (1.4.1) holds for any word w in place
of 5. Let w = aY ... а„(а{ e A). Then, since dp is an homomorphism,
<5P(ai  a„) = ^p(a1).. .dp(a„)
= П (ai ® ®	®	® ai ® ’ ’ ’
i = 1
® 1 + •  • + 1 ® 1 ®   -® at)
E (О®- -® (w|/p)
{1....»| = /|U •   <j/p(disjoint)
=	£	(w, UY LU- • -LU Up)Ui ® Up,
ui....upeA*
the last equality by definition of the shuffle product.	□
The shuffle product is sometimes defined recursively in the following
asymmetric way: 1 ш w = w ш 1 = w for any word w, and if и = au',
1 Lie polynomials
26
v = bv'(a, b e A, u', v'eA*), then
(au') lli (bv') = a(u' lli (bv')) + b((au') ш v').
(1-4.2)
This definition is equivalent to the previous one: indeed, a shuffle of au' and
bv' is either an a followed by a shuffle of u' and bv', or a b followed by a
shuffle of au' and v'.
Equation (1.4.2) expresses the fact that a certain mapping is a derivation
of the shuffle algebra К<<Л>>. Indeed, fix a letter a and define a continuous
linear endomorphism S^a~1S of К<<Л>> by its effect on each word:
a~Yw = и if w = au and ue A*, a~Yw = 0 if w does not begin with a. Then
a xS is a derivation of the shuffle algebra, i.e.
ан(5ш T) = (а~15)шТ + Хш(а'Т).	(1.4.3)
Indeed, eqn (1.4.3) follows easily from eqn (1.4.2).
1.5 DUALITY CONCATENATION/SHUFFLE
Recall that there is a canonical scalar product
к<л> x к<л> к, (p,e)= e (f. »xe. ")
weA*
In other words, it is the unique scalar product on K(A) for which A* is
an orthonormal basis. Similarly, one defines on К<Л>®Р the scalar product
for which
{«i ®	® up, Up ..., up 6 A*}
is an orthonormal basis.
Let us view the shuffle product as a linear mapping
sh: K(A) ® K<(A> -> K(A>, и ® v -> и ш v.
This mapping is homogeneous and degree-preserving for the canonical
degree on К<Л> ® К<Л>, i.e. deg(u ® v) = deg(u) + deg(v) = |u| + |v| for
any words u, v.
Consider the adjoint (for the above scalar products) of the mapping sh,
that is the unique mapping sh*: K(A) -> К<Л> ® К<Л> such that for any
polynomials P, Q, R
(sh(P ® Q), R) = (P® Q, sh*(R)).
Note that by Proposition 1.8, we have
b(R) = b2(R) = £ (R, и ш v)u ® v.
u,veA*
1.5 Duality concatenation/shuffle
Thus, for any words u, v
27
(sh(u ® v), R) = (w lli v, R) = (w ® v, d(R)).
Hence, by linearity, for any polynomials P, Q, R
(sh(P®Q), R) = (P®Q, <5(R))-
In other words, d is the adjoint of sh. Note that by definition, d is a
homomorphism for the concatenation structure (see Section 1.3). We obtain
a similar result if we interchange concatenation and shuffle. Indeed, denote
by д' the adjoint of the concatenation product, viewed as a linear mapping
cone:	® K(A) -> K(A), u®vi—>uv.
Thus д' = cone*. We have for any polynomials P, Q, R
(PQ, R) = (conc(P ® Q), R) = (P ® Q, conc*(R)) = (P ® Q, d'(R)).
This implies that for any word w
d'(w) = E (u ® v’ d'(w))u ® v
u, ve A*
= £ (uv, w)u ® V
= £ u®v.
w= uv
This formula—which completely describes д'—means, informally, that d'(w)
is equal to all decompositions in two words with respect to the concatenation
product. Similarly, <5(w) is equal to all decompositions in two words with
respect to the shuffle structure, with multiplicities. Define a linear mapping
e: K(A> - K, e(P) = (P, 1).
We have partially proved the following result.
Proposition 1.9 The adjoint d of the shuffle product is a homomorphism for
the concatenation product. The adjoint д' of the concatenation product is a
homomorphism for the shuffle product. The mappings d and д' are described by
d(w) = £ (w, и ш v)u ® v,	(1.5.1)
u, ve A*
d'(w) = £ (w, uv)u®v.	(1.5.2)
u, ve A*
Proposition 1.9 implies that there are two bialgebra structures on К<Л>: one
with the concatenation product and d as coproduct, the other with the shuffle
product and д' as coproduct.
28	1 Lie polynomials
Proof It is enough to show that д': K(A) -> К<Л> ® K(A) is a shuffle
homomorphism. Let us write that d is a concatenation homomorphism, i.e.
5(xy) = <5(x) <5(y). By eqn (1.5.1), this means that
£ (xy, U LU v)u ® V = I £ (x, Ui LU fJUi ® Vi jl £ (y, U2 Ш V2)u2 ® V2 I
U,V	\ut,Vt	J\U2,V2	/
= J (x,ulmvl)(y,u2mv2)(u1u2®v1v2).
U2.V2
Hence, for any words x, y, u, v, by taking the scalar product of both sides
with и ® v we obtain
(xy, wujv)= Y (x,Ui ^nvl)(y,u2mv2)(u1u2,u)(v1v2,v).
Ui , V] , U2, V2
Note that this formula has a complete symmetry concatenation/shuffle. We
just have to reverse the computation, interchanging shuffle and concatena-
tion. Thus, by (1.5.2) and linearity,
д'(и ш v) = £ (w ш v, xy)x ® у
х.У
=	£ (Wj Ш Up x)(l42 Ш V2, y)(u, UiU^V, v1v2)x ® у
X, у, ui, U2,	, V2
=	£	(u, UjUjXv, ViV2)l £(Uj LU vl,x)x\®l £(m2lu v2, y)yj
Wl,U2,Vi,U2	\ X	/	\ У	/
=	£	(u,u1u2)(v,v1v2)(u1wv1)®(u2ujv2)
Ul,U2,Vi,V2
= 1 £ (w,	® U2 j Ш I Y, (Г’ yly2)yl ® V2 )
\W]*W2	/	/
= д'(и) ш <5'(0>
where of course K(A) ® К (A) has the shuffle structure defined by
(«! ® U2) LU (Vj ® V2) = (Ui Ш Vj) ® (u2 Ш v2).	□
We define now on End(K</l», the К-module of linear endomorphisms
of К (A), a K-algebra structure, different from its usual algebra structure
defined by composition of endomorphisms. For f, g in End(K</l», define
f *g in End(K</l» by the formula
f *g = conc-^ {f ® g)° d.	(1.5.3)
In other words, for any polynomial P, one has by Proposition 1.8:
X <P,u4iv)f(u)g(v).	(1.5.4)
u, ve A*
We call the product * the convolution. Recall that the endomorphism
а: К < A) -> К (A) sends each word w = ax ... an (a{ g A) on its signed rever-
1.5 Duality concatenation/shuffle	29
sal a(w) = (— l)"a„... Now define an associative algebra j/, as the
complete tensor product
= К<Л>® К<Л>,
with the shuffle product on the left of ®, and the concatenation on the right
of ®. More formally, an element of si is an infinite linear combination
au,rM ® v and the product in si is defined by
X ct„fVu® V V X Px,yx®y I = X iuJxfumx)®(vy).
u,v	/ \x,y	J u,v,x,y
Note that each endomorphism f of К<Л> is completely described by its
canonical image £иеЛ* и ® /(w) in si. Note also that the identiy endo-
morphism is mapped onto £u и ® u.
Proposition 1Л0 With the convolution, End(K<4>) acquires an associative
algebra structure, whose unit element is e. The inverse in this algebra of the
identity of KfA") is a, which is an anti-automorphism of K(A)> for the
concatenation product, and an automorphism for the shuffle product. The
canonical embedding End(K<^>) -> si, f i—► £u и ® /(u), is an algebra homo-
morphism for this product. In other words
E w® ((/*#)(w)) =	/(M) r ® #00 )	C1-5-5)
w	\ u	J \ V	J
where the product is taken in si (shuffle on the left of ®, concatenation on the
right).
This proposition implies in particular that К<Л> has a Hopf algebra
structure, with concatenation as product, 5 as coproduct, and a as antipode.
Proof We start with eqn (1.5.5). The right-hand side is
£ (w ш v) ® (f(u)g(v)) = £	(w, и ш v)w ® (f(u)g(v))j
U,V	U, V \ w	/
= E w ® (E u ш OM») )•
w	\u,v	/
Hence, eqn (1.5.5) is equivalent to
(/ *0)(w) = £ (w, и ш v)f(u)g(v),
U, V
which is the definition of f *g (see eqn (1.5.4)). Since the product in .я/ is
associative and since
/h->£u®/(u),	End(K<^>) -> si
30	1 Lie polynomials
is injective, the convolution is also associative. Moreover, the image of e is
1(8)1, which is the neutral element of stf. Hence, e is the neutral element for
the convolution. The fact that a is an automorphism of the shuffle algebra
(and an anti-automorphism of the concatenation algebra) is clear by
inspection.
Now, by Lemma 1.5, we have for any polynomial P, conc°<5(P) = (P, 1),
where <5 = (id ® a) ° b. In other words, by eqn (1.5.3), we have id * a = e. So
a is the right inverse of id for the convolution. This may be written as
<x(u) j = 1 ® 1.
v	/ \ и	/
Now, a is an automorphism for the shuffle, hence a ® id is an automorphism
of . Applying a ® id to the last equation, we obtain
£ a(v) ® v Y £ a(u) ® a(u) j = 1 ® 1.
Now, £u a(w) ® a(w) = Lu и ® и and a(v) ® v = v ® a(v) because a is
an involution. Hence (£ v ® a(v))(£ и ® u) = 1 ® 1, which means that a is
also a left inverse of id.	□
If h is an endomorphism of the concatenation algebra K<A>, we say that
h is a Lie endomorphism if h(A) У2(Т), or equivalently, if	^f(A).
This is particularly the case if h is homogeneous, i.e. h preserves homogeneous
polynomials and degrees; equivalently, h(a) is a homogeneous polynomial of
degree 1 for each letter a; a particular case of this is when h is a substitution
of letters, i.e. h(A) A.
For later use, we prove the following lemma.
Lemma 1.11 If f, g 6 End(K</l» commute with a Lie endomorphism h, then
so does f *g.
Proof Since h(a) is a Lie polynomial for any letter a, we have by Theorem
1.4(iii)
3 -> h(a) = h(a) ® 1 + 1 ® h(a).
Furthermore,
(h ® h) о 3(a) = (h® h)(a ® 1 + 1 ® a)
= h(a) ® 1 + 1 ® h(a),
since /1(1) = 1. Thus 3 h = (h ® h) 6, both being concatenation
1.5 Duality concatenation/shuffle	31
homomorphisms. Hence, by (1.5.3),
(/ * g) ° h = cone ° (/ ® g) ° 3 ° h
= cone ° (/ ® g) ° (h ® h) ° 3
= cone ° ((/ ° h) ® (g о h)) ° 3
= cone ° ((/i ° /) ® (h ° g)) ° 3 (by hypothesis)
= cone ° (h ® h) ° (/ ® g) ° 3
= h° cone °(f ®g)°3
= h°(f*g),
because cone ° (h ® h) = h ° cone, h being a concatenation homomorphism.
□
There is a simple extension ofeqn (1.5.3). As in Section 1.4 we shall denote
by 3p the concatenation homomorphism
K<A> -
such that for any letter a, 3p(a) = a ® 1 ®	® 1 + 1 ® a ®	® 1 +  • • +
1 ® 1 ®	® a. It is easy to establish the following slight extension of
Theorem 1.4(iii): if P is a Lie polynomial, then
3p(P) = P®1®--®1 + 1®P®---®1 + -- -+1®1®---®P.
(1.5.6)
Moreover, for any endomorphisms /15..., fp, their convolution is
/i*- • •*/₽ = Сопере/! ®- • ®fp)°3p,	(1.5.7)
where concp(w1 ® • •  ® up) = ur... up. This may be seen for example by
expressing the product j\ * • • • * fp in the algebra stf and using Proposition
1.10.
Because of the symmetry concatenation/shuffle in the algebra stf, there is
another Hopf algebra structure K(A), with shuffle as product, 3' as
coproduct and the same antipode; the convolution *' in this case is defined by
f *' g = sh°(f ®g)°3',
and it corresponds to the product in л/, under the embedding
End(K<A>) i—► .$/,
ve A*
Furthermore, define 3P and shp by
<5p(w) = X	shp(ul ® • • -® Up) = ur LU- • -Ш Up.
W = U 1 . . .Up
32	1 Lie polynomials
The mapping b'p is a shuffle homomorphism, and we have
/1 *'. .. *' fp = Shp о (Л ® • •  ® 4) о 4,	(1.5.8)
for any endomorphisms ., fp of К<Л>.
It may be worthwhile to note that if f, f* are adjoint endomorphisms,
i.e. (/(w), v) = (u, /*(v)) for any words u, v, then one has in d
Yu® = Yf*(v)®v	(1.5.9)
Conversely, this equality implies that f and f* are adjoint.
The foregoing results have as applications some combinatorial identities
on words. By Lemma 1.5, we have the identity A ° <5(P) = r(P) for any
polynomial P, where A, J are the linear mappings defined by: D(w) = |w|w
for any word w, z(P ® Q) = D(P)Q, <5 = (id ® a) ° 5, and r(«x ...«„) =
[als..., [a„_i, a„] ...] for any letters аг,...,а„ (Lie bracketing from right
to left). We may write A = cone ° (D ® id), hence the above identity may be
rewritten as:
r = cone ° (D ® id) ° (id ® a) ° <5 = cone ° (D ® a) ° d = D*tx.
Since id is the inverse of a for convolution, we obtain: r * id = D. We write
the latter equality in the algebra si with the help of eqn (1.5.5), and use the
fact that if p denotes the adjoint endomorphism of r, then by eqn (1.5.9):
£ w® r(u) = Yp(v)®v-
U	V
Thus
£ |w|w ® w = £ w ® D(w) = I £ p(v) ® v II £ и ® и I
w	w	\ V	J \ U	J
= s (p(v) ш M) ® vu-
U, v
Hence, we deduce
£ p(v) ш и = |w|w
w = vu
for any word w. It is interesting to note that p has an effective recursive
definition
p(l) = 0, p(a) = a ifaeA,	(1.5.10)
p(aub) = ap(ub) — bp(au) if и e A* and a, b e A.
Indeed, since p and r are adjoint we have for any words v, w
(r(v), w) = (v, p(w)).
(1.5.11)
1.6 Appendix
33
It is enough to check that p defined by (1.5.10) satisfies (1.5.11). Note first
that p(w) and r(w) are homogeneous polynomials of degree |w|, equal to 0
if w = 1, equal to a if w is the letter a. Hence (1.5.11) is true for |w| = 0, 1.
Note also that (1.5.11) is true if |v| / |w| since in that case both sides vanish.
So we need only consider the case |v| = |w|. We shall use the following identity
(xy, zt) = (x, z)(y, t) if x,y, z, te A*, |x| = |z|, |y| = |t|.	(1.5.12)
Suppose now |w| > 2, hence w = aub with a, b in A. Then by (1.5.10), the
right-hand side of (1.5.11) is (v, p(w)) = (v,ap(ub)) — (v,bp(au)). Since
|v| = |w|, we have v = ex, c g A, x g A*. Then by (1.5.12), (v, p(w)) =
(c, a)(x, p(ubj) — (c, b)(x, p(au)). By induction, this is equal to (c, n)(r(x), ub) —
(r(x), au)(c, b). This in turn is equal, by (1.5.12), to (cr(x), aub) - r(x)c, aub) =
(cr(x) — r(x)c, aub) = (r(v), w), which completes the proof of (1.5.11).
We can thus state the following result.
Theorem 1.12 Define the linear endomorphism p of К<Л) by (1.5.10), and
the endomorphism r by r(aY ...«„) = [a15..., [a„_ 15 «„] ...] for any letters
аг,... ,a„. Then r and p are adjoint endomorphisms of К<Л>, and for any
word w, one has
£ p(v) lli и = |w|w,	(1.5.13)
vv = vu
or equivalently, in the convolution algebra K^A), r*id = D, where
D(at ...a„) = naY ... an.
1.6 APPENDIX
1.6.1 Support of the free Lie algebra
We call support of the free Lie algebra the subset 5 of A* consisting of
those words which appear (with a nonzero coefficient) in some Lie poly-
nomial over Z. It was shown by Duchamp and Thibon (1989) that a word
w is in Л*\5 if and only if w is either of the form an (ae A, n> 2), or a
palindrome (i.e. a word equal to its reversal) of even length. One part of
this result goes as follows: if w = an (a g A, n > 2), then by projection
&(A) -> ^(a) = T.a, one sees that w is not in 5; if w is a palindrome of even
length, then a(w) = w, and a(P) = - P for each Lie polynomial by Lemma
1.7, so that w has zero coefficient in P, and w is not in 5. In order to show
that each word not of the previous form is in 5, the authors construct an ad
hoc family of Lie polynomials.
A more general problem is the following: define for each word w, the ideal
34	1 Lie polynomials
(n) of Z consisting of the coefficients of w in all Lie polynomials over Z. The
previous result characterizes those words with n = 0. Which words have
n = 1? (This problem was posed by M.-P. Schiitzenberger.)
1.6.2 Formulas in characteristic p
For any letters a, b, and integer n, one has the identity
ad(a)n(b) = [a, [a,..., [a, b] ...]] = £ ( " )( - 1)W
i + j = n \ I /
If К is of prime characteristic p, one deduces, for any n > 0,
ad(a)pn(b) = [u<b],	(1.6.1)
and
od(a)p-1(b) =	£ a‘baj.	(1.6.2)
i+j=p-1
This is because, in characteristic p, one has (p") = 0 for i = 1,..., p" — 1,
(p;1) = (-1)* for / = l,...,p- 1 and(—l)p-1 = 1.
Formula (1.6.1) implies that for any n, m, one has
[a"", b₽m] = ad(a)pn(bpm)
= ad(a)pn~1 ° ad(a)(bpm)
= ad(a)p”~\[a, bpm])
= —ad(a)pn~1([bpm,a'])
= - ad(d)pn ~1 о ad(by”\a).	(1.6.3)
Define a polynomial Ap(«, b) by the formula
(a + b)p = ap + bp + Ap(a, b).	(1.6.4)
Then Ap(a, b) is a Lie polynomial (in characteristic p). Indeed, by definition,
Ap(a, b) is the sum of all words of length p on the letters a, b, except the two
powers ap, bp. Hence
Ap(a,b) = £ Ap k(a, b),
к = 1
where Ap k is the sum of all words of length p and partial degree к in b. So
it suffices to show that each Apk is a Lie polynomial. Observe that
£ kAPik(u, b) = (a + ь?ь(а + bY-
к=1	i + j = p— 1
1.6 Appendix	35
Since 1,..., p — 1 are invertible in characteristic p, and since the free Lie
algebra is finely homogeneous by Lemma 1.3(i) it is enough to show that
the right member is a Lie polynomial. But it is equal to ad(a + b)p~ \b), by
(1.6.2).
The previous formulas are called the Jacobson formulas (Jacobson 1937;
see also Zassenhaus 1939).
1.6.3 Free Lie p-algebra
Let К be of prime chracterisitic p and denote by J?P(A) the К-subspace of
К<Л> generated by the p"th powers of Lie polynomials (n > 0). By (1.6.3),
the Lie bracket of two elements of ^fp(A) lies in ^(A). This shows that
Ур(А) is a Lie subalgebra of К<Л> containing ^(A), called the free Lie
p-algebra. See Section 2.5.2 for a further study.
1.6.4 Identities on words
The fact that a is the antipode for the two Hopf algebra structures on К<Л>,
i.e. that £ и ® a(w) = £ a(v) ® v is the inverse of £ w ® w in the algebra 3/,
may be expressed in each of the four following equivalent identities on words
(w is a nonempty word):
0 = £ (— l)|u|u lu v
w = uv
= £ (— l)|l,|w ш v
w~uv
= £ (— l)|u|(w, и ш v)Uv
U, V
= £ (— l)|,?l(w, и Ш v)uv,
U, V
where u denotes the reversal of и (see Schmidt 1990).
1.6.5 Derivations
Theorem 1.4(iii) means that the primitive elements for the coproduct 6 are
the Lie polynomials. If P is a Lie polynomial, then the adjoint of the left
multiplication by P (in the concatenation algebra) is a derivation in the
shuffle algebra. This is a consequence of (1.4.3) and the fact that the Lie
bracket of two derivations is a derivation. Conversely, if the adjoint of the
left multiplication by P is a derivation in the shuffle algebra, then P is a Lie
polynomial. For the proof, use Theorem 1.4(iii). Similar results hold of course
for the adjoint of the right multiplication.
36	1 Lie polynomials
Symmetrically, the primitive elements for the coproduct д' are the linear
combination of letters. If P is such a polynomial, then the adjoint of the
shuffle product by P is a derivation for the concatenation structure. If P = a
is a letter, it is the usual derivation d/да which maps each word w on
£w=uei? uv- & has characteristic zero, the intersection of the kernels of all
these derivations, for a in A, is equal to the (concatenation) subalgebra M
of К<Л> generated by the Lie polynomials without linear part: indeed, it is
easy to see that the derivative of each element of M vanishes; conversely,
each element of К (A) has a unique expansion
p = E p. П
a aeA
where the sum is over all a in N(j4), where Pa 6 M, and where the product is
taken in decreasing order, for some total order on A (because К (A) is the
enveloping algebra of &(A); see Theorem 0.5 and Theorem 0.2); if dIdaP = 0,
for any letter a, then it is easy to see that all Pa = 0, except Po. For related
problems, see Lenormand (1969/70).
1.6.6 An identity of Baker (1905)
Let /: К<Л> -> У(А) be the ‘Lie bracketing from left to right’, i.e. the linear
function such that /(1) = 0, 1(a) = a for any letter a, and l(Pa) = [l(P), d] for
any polynomial P. Then, for any polynomials P, Q
/(W)) = m/(0].	(1-6.5)
It is enough to check this identity when P, Q are words; then, a few lines of
computations and an induction on the length of Q give the result.
From (1.6.5), one deduces that
l(U(P), ПОЛ) = U2(P), 1(Q)1 + U(P), Z2«2)J -	(1.6.6)
By Jacobi’s identity, each Lie polynomial is of the form l(P) (see Section
0.4.1). Thus (1.6.6) implies that 1\.У(А) is a derivation of Lie algebra, that is
/([Л QD = W>), QI + [Л №)],	(1.6.7)
for any Lie polynomials P and Q. Now, (1.6.7) implies easily, by induction
on n, that for each homogeneous Lie polynomial P of degree n, one has
l(P) = nP.
This is Theorem 1.4(v).
1.6 Appendix
37
1.6.7 Kernel of the left to right bracketing
The kernel of the linear mapping /: К<Л> -> У2(Л) is the right ideal of К<Л>
(concatenation algebra) generated by the polynomials
P/(P), РеК<Л>.	(1.6.8)
This was proved by Cohn (1951). We outline a proof which works only
in characteristic 0. Observe that Ker / is a right ideal: this is because
l{Pd) = [/(P), a] for any letter a. Now, Baker’s identity (1.6.5) shows that
each polynomial of the form (1.6.8) is in Ker /.
For the converse, we use the identity
nP = у (P, и lu v)ul(v)	(1.6.9)
u, veA*
for any homogeneous polynomial of degree n (proof below).
Identity (1.6.9) may be rewritten
(n - 1)P = Z(P) + У (P, и ш v)ul(v)
u,v # 1
= /(P) + У (P, U LU u)ul(u) + У (P, U LU v)ul(v).
U # 1	U # V
U, V # 1
The second summation is a linear combination of ul(y) + vl(u), because the
shuffle product is commutative. The latter polynomial is equal to
(u + v)l(u + v) — ul(u) — vl(y).
Thus, if P is in Ker /, it is a linear combination of polynomials of the form
(1.6.8).
To prove identity (1.6.9), we use the identity r*id = D of Theorem 1.12.
By symmetry, we have id* / = D, which by (1.5.5) may be rewritten in the
algebra за/
у и ® V ® /(r)^ = y |w|w®w.
Recall that за/ is the complete tensor product of the shuffle algebra by the
concatenation algebra. Thus we obtain for any word w
|w|w = У (w, U LU v)ul(v).
U, V
This implies (1.6.9), by linearity. For related work, see Labute (1978) and
Patsourakos (1987).
38
1 Lie polynomials
1.6.8 Sweedler dual coalgebra
The linear mapping д' of Section 1.5 is extended to formal series by the
formula
d'(S) = £ (S, uv)u ® v.
U, V
Hence, the image d'(S) is an element of the complete tensor product
К<<Л» ® К«Л», of the form
i'(S) = £ S, ® Tt.	(1.6.10)
i
The dual coalgebra (see Sweedler 1969, Chapter VI) of the concatenation
algebra К<Л> is by definition the set of formal series (identified as in Section
1.1 with linear functions on such that the sum (2.6.10) is finite;
that is, the series 5 such that d'(S) actually lies in the tensor product
К<<Л» ® К«Л>>. Denote by R this set of series. Since д' is a homomor-
phism for the shuffle product (Proposition 1.9), R is closed under shuffle
product. Hence R becomes a bialgebra, with shuffle product and д' as
coproduct. It is easily verified that a(R) cz R, hence R is actually a Hopf
algebra.
It is interesting to note that these series occur in a quite different context,
derived from automata theory. It is known, and easily shown, that condition
(1.6.10), with a finite sum, is equivalent to the fact that the kernel of S,
considered as a linear form on K(A), contains an ideal (concatenation
structure) of К<Л> of finite codimension (we assume for simplicity that К
is a field; this ideal may be equivalently left, right or two-sided). This
condition is also known in automata theory, where such a series 5 is called
recognizable.
These series have an equivalent definition: indeed, by the Kleene-
Schiitzenberger theorem, a series is recognizable if and only if it is rational, i.e.
belongs to the smallest subalgebra of	(concatenation structure),
which contains the polynomials, and which contains the inverse of all its
invertible elements.
One connection with automata is the following result of Schiitzenberger
(1961): a language L (i.e. a subset of A*) is recognizable by a finite automaton
if and only if its characteristic series is rational (or recognizable). References
on this subject are Eilenberg (1974), Salomaa and Soittola (1978), Lallement
(1979), and Berstel and Reutenauer (1988).
Similarly, one may raise the question which are the series 5бК((Л))
such that
Ж)6К«/1»®К«Л».	(1.6.11)
Among these series are the Lie series (e.g. the Hausdorff series) and the
1.7 Notes
39
exponentials of Lie series (see Section 3.1). Moreover, the set of series
satisfying (1.6.11) is a concatenation subalgebra of K<<4>>, because 3 is a
homomorphism. If К is an algebraically field of characteristic 0, then the
converse holds: the set of series satisfying (1.6.11) is the concatenation
subalgebra generated by the Lie series and their exponentials. This is a
consequence of Sweedier (1969, Section 13.1 together with Lemma 8.0.l.c
and Section 7.2). Note that in the case of a one-letter alphabet, this result
expresses the well-known fact that a series £ antn satisfies a linear differential
equation with constant coefficients if and only if it belongs to the subalgebra
°f К [[t]] generated by t and the series e°" (a e К).
1.7 NOTES
Theorem 1.4 characterizing Lie polynomials has different sources: charac-
terization (ii) is from Finkelstein (1955), (iii) is due to Friedrichs (1953)
and proved by Cohn (1954), Magnus (1953, 1954), Lyndon (1955b), and
Finkelstein (1955), (iv) is from von Waldenfels (1966b), and (v) is due to
Dynkin (1947), Specht (1948), and Wever (1949). For the proof, we have
followed von Waldenfels (1966b), who establishes Lemma 1.5. Other proofs
may be found in Ree (1958) (his proof rests essentially on eqn (1.5.13)), and
Garsia (1990); see also Section 8.6.5.
The introduction of the shuffle product for the study of Lie polynomials
is due to Ree (1958). A signed shuffle product appears in earlier papers of
MacLane (1950) and of Eilenberg and MacLane (1953). Today, the shuffle
product is well understood in the light of Hopf algebras. An analogue of this
product is defined more generally on the dual of each enveloping algebra
(Dixmier 1974): it is simply the adjoint of the coproduct 3, defined for any
Lie element as here by <5(P) = P ® 1 + 1 ® P.
Proposition 1.9 is a particular case of a result in bialgebras: if an associative
algebra has a coproduct which is an homomorphism, then its product is a
coalgebra homomorphism (see Abe 1980, Theorem 2.1.1). The fact that 3’ is
a shuffle homomorphism was also noted by Chen (1968, Theorem 1.8). The
convolution product * in End(K<4>) is defined more generally in any
bialgebra. For more on coalgebras, bialgebras and Hopf algebras, see Milnor
and Moore (1965), Sweedler (1969), Bourbaki (1972), Abe (1980), and
Hochschild (1981). Theorem 1.12 is due to Ree (1958). This result and other
identities of the same kind were rediscovered and applied in control theory
by Crouch and Lamnabhi-Lagarrigue (1989); one reason for this is that the
shuffle product corresponds to the product of iterated integrals (Chen 1957;
Fliess 1981; see also Section 6.5.4).
2
Algebraic properties
In Section 2.1, we introduce the weak algorithm for noncommutative
polynomials, and a similar tool for Lie polynomials. In Section 2.2, we prove
the theorem of Shirshov-Witt that each Lie subalgebra of a free Lie algebra is
free. In the next section, we show that the automorphism group of JT(A) is
generated by elementary automorphisms, and give a jacobian-like condition
characterizing automorphisms. In the final section, we characterize free sets
of Lie polynomials and prove the defect theorem. In the appendix, we discuss
uniqueness of rank and restricted Lie algebras in characteristic p.
In this chapter, К is a field.
2.1 THE WEAK ALGORITHM
Cohn’s weak algorithm is an extension to noncommutative polynomials of
the Euclidean algorithm. We say that a finite family P1;..., Pn of polynomials
in K(A > is (right) dependent if either some Pj = 0 or if there exist polynomials
Qi,..., Qn such that deg(£7 PjQj) < maxJ(deg(PJQJ)). Observe that if non-
zero polynomials P1;..., Pn are right К<Л>-1теаг dependent, then they are
dependent. Further, a polynomial P is (right) dependent on Pt,..., Pn if either
P = 0 or if there exist polynomials Q1;..., Qn such that
deg(P - X PjQj\ < deg(P),	(2.1.1)
and for j = 1,..., n
<teg(PjQj) < deg(P).
The next result is due to Cohn (1961).
Theorem 2.1 Let Pt,..., Pn be a dependent family of polynomials with
deg(Pj) < • • • < deg(P„). Then some Pt is dependent on Ръ..., P^.
For the proof of Theorem 2.1 we need a linear operator on К<Л) (symmetric
to the operator introduced at the end of Section 1.4). Let и be any word.
41
2.1 The weak algorithm
Then define the polynomial Pw-1 by
Pn 1 = S (P, wu)w.
we A*
It is clear that P i—► PtT1 is a linear endomorphism of К<Л>; if P = w is a
word, then wu~1 = x if w = xu, and wu~1 = 0 if w does not have the suffix u.
The following relations hold:
deg(Pw "1) < deg(P) - |u|,	(2.1.2)
P(ur)1 = (Pr^u1.	(2.1.3)
Moreover, for any letter a
(PQ)a-1 = P(Qa"1) + (Q, l^a"1,	(2.1.4)
where (Q, 1) is as usual the constant term of Q. All these relations are easily
verified when P, Q are words, and then extended by linearity.
Lemma 2.2 If P, Q are polynomials and w is a word, then there exists a
polynomial P' such that
(PQ)w~1 = P(Qw^ + P',
and P = P' = 0 or deg(P') < deg(P).
Proof We may assume that P 0. If w = 1, then the lemma is evident,
because P i—► Pw~1 is then the identity.
Otherwise, let w = au, for some letter . Then we have by induction
(PQ)u-1 = P(Qw"1) + P', deg(P') < deg(P).
By (2.1.3), we have
(PQ)w-1 =((PQ)w-1)a~1 =(P(Qu-1))a-1 +P'a~1.
Hence, by (2.1.4) and (2.1.3)
(PQ)w1 = P((Qw-1)a-1) + (Qm1. l)Pa-1 + P'a-1
= P(Qw~1) + P",
where P" = (Qw-1, l)Pa“1 + P'a-1. By (2.1.2), deg(P") < deg(P), which
proves the lemma.	□
Proof of Theorem 2.1 We may suppose that no P, = 0. Hence deg(£, ^Q,) <
maxi(deg(^Q,)) for some polynomials Q1;..., Q„. Let r = max(deg(^Q,))
and I = {i|deg(^Qi) = r}. Then R = Xiei PiQihas degree < r. Let к = sup(/);
42
2 Algebraic properties
then i e I => deg(f,) < deg(Pk). Let w be word such that |w| = deg(Qk)
and 0 / (Qk, w) = a-1 e K: this is possible, because Qk T- 0 (otherwise
deg(P) < r = deg(PkQk) = — oo, which is not possible).
By Lemma 2.2 we have
iel	iel
for some polynomials P't with degiP'J < deg(^). Since 1 = a*", we have
A + « E	=	(2.1.5)
ie/\k	iel
Now, by (2.1.2)
deg(Pw-1) < deg(P) - |w| < r - |w| = deg(PkQk) - deg(Qk) = deg(Pk).
Furthermore, deg(P-) < deg(Pf) < deg(Pk). Hence, the degree of the right-
hand side of (2.1.5) is <deg(Pk). Moreover, by (2.1.2) we have for every i e I:
deg(^(Q,w_1)) = deg( Pf) + deg(Qfw-1) < deg(^) + deg(Q,) - deg(Qk)
= r - deg(Qk) = deg(Pk).
This shows that Pk is dependent on Ph i e I \k; a fortiori, Pk is dependent on
Рр-.-’Л-Р	□
We say that a Lie polynomial P is Lie-dependent on Lie polynomials
Рг, . . . , Pn if P = 0 or if there exists a Lie polynomial f(x1,...,x„) in
..., xn) such that deg(P — /(P1;..., P„)) < deg(P), and that each word
in the X( appearing in f is of some degree d, in with £ d{ deg(/<) < deg(P).
Theorem 2.3 Let Pt,..., Pn be a dependent family of Lie polynomials, with
deg(Pj) < • • • < deg(P„). Then some P( is Lie-dependent on Рг,..., Р(_г.
We need the following result. It is valid mutatis mutandis in any enveloping
algebra.
Lemma 2.4 Let LT be a Lie subalgebra of LT(A). Then each Lie polynomial
which is in the right ideal of К {A} generated by LT is already in LT.
Proof Let LA be a totally ordered basis of LT {A), containing a basis of LT,
and such that
P, Q&LA. PeLT. Q$LT => P >Q.	(2.1.6)
We claim that the set of polynomials of the form
Л ...P„, n > 1,Ле< Л Л > • • • > P„,	(2.1.7)
2.1 The weak algorithm	43
linearly generates the right ideal I of К<Л> generated by <£. Indeed, by
Theorem 0.5 К<Л> is the enveloping algebra of ^(Л), so that by Theorem
0.2,1 is linearly generated by the polynomials
P =	... Pn, n > Ute.A Pt	(2.1.8)
So it is enough to show that each polynomial (2.1.8) is a linear combination
of polynomials (2.1.7); we do this by induction on (n, t(P)), ordered lexico-
graphically, where t(P) is the number of (i, j), 1 < i < j < n, such that P{ < Pj.
If n = 1, then P is of the form (2.1.7), and there is nothing to prove. If n > 2,
and t(P) = 0, then P again is of the form (2.1.7). Hence, we may assume
t(P) > 1: then there exists i such that P( < Pi+ P If i = 1, since Pt e .У. we
deduce by (2.1.6) that P2 e ЛР. Then [РрРгЗб^, because is a Lie
subalgebra, hence [P1; P2] = ayQXQj e <£ n ccj e K). We have
P = 1Л Л1Л • • • + P2PtP3 ... P„
=	+	- р,-
j
Polynomial QjP3 ... P„ is of the form (2.1.8), with a smaller n; moreover,
P2PiP3 ... P„ is also of the form (2.1.8), with the same n and smaller t(P).
Hence, by induction, all these polynomials are linear combinations of
polynomials of the form (2.1.7), and so is P. Suppose now that i > 2. Then,
we have [/>. Pi+ J = flkRk (Rk e <£), and
P = Л • • • Pi- 10Л Pi+ il+Pi+ iPi)Pi+2 - Pn
= X PkPl • • • Pi-lPkPi+ 2 • • • Pn + Pl • • • Pi- 1^ +	2 • • • Pn-
к
Then, a similar argument shows by induction that P is a linear combination
of polynomials of the form (2.1.7). This proves the claim.
Now, let К be a Lie polynomial which is in I. Then we may write
*=Z«ce=ZM’	(2.1.9)
Qt.H	P
where aQ, fiP are in K, and where the second summation is over polynomials
P of the form (2.1.7). Now, the decreasing products	2,6^,
Qi > • • • > Qn, are linearly independent (Theorem 0.2). Hence, (2.1.9) implies
that К is a linear combination of polynomial Q e & n In particular, R
is in %?.	□
Denote by P the highest homogeneous component of a polynomial P, with
P = 0 if P = 0.
Proof of Theorem 2.3 We may suppose that no Pf = 0. By Theorem
2.1, some P, is dependent on P^..., Р^. In other words, we have
44	2 Algebraic properties
<teg(pi	PjQj) < de8 for some polynomials Q7, and deg(PyQ?.) <
deg(PjQj) for j = 1,..., i — 1.
We deduce that
P.-UA,
where the sum is over those j such that 1 < j < i and deg(PjQj) = deg (Pf).
Hence P( is in the right ideal of К (A) generated by P1;..., P(_ P Observe
that these polynomials are Lie polynomials, by Lemma 1.3(i). This implies
by Lemma 2.4, applied to the Lie subalgebra generated by P1;...,	1; that
Pi is in this subalgebra.
We have Pt = д(Ръ ..., Pt- J for some Lie polynomial g in .y'f.Xj...., J.
Let Pj = deg(Pj), p = deg(Z-). Let g = £(<0 g(d), where the sum is over all
(i - l)-tuples (d) = (d^ .. and where g(d} is the homogeneous compo-
nent of g of degree dj in Xj, j = 1,..., i — 1. Let f = g(d} where the sum
is over all (d) with djPj = p. By homogeneity, we have
Pi = f (Pi,..., P(-J.
Moreover, £P(xr,..., x,_ J is homogeneous (Lemma L3(i)), so that each gid}
is a Lie polynomial, hence f is too. Now, by construction, f(Pr,..., Pi-f)
is equal to f(Pt,..., Р(-г) plus a polynomial of degree less than p. This
shows that P( is Lie-dependent on P1; P	□
2.2 SUBALGEBRAS
In this section and the following, К is a field. The next result is due to
Shirshov (1953) and Witt (1953, 1956).
Theorem 2.5 Each Lie subalgebra of a free Lie algebra is free.
Proof Let J*9 be a Lie subalgebra of the free Lie algebra £P(A). Denote by
En the subspace
E„ = {Pe^|deg(P)<n}.
Let <E> denote the Lie subalgebra generated by £c £P(A). Moreover, let
E'„ be the subspace of En defined by
E'n = En и <E„-i>-
We have of course
{0} = Eo = E\ s £1 s E'2 S E2 S    S £„_ , £ E; s £„ s • • •.
Let X„ be a subset of E„ which defines a basis of £„ mod E'„. Define
2.3 Automorphisms	45
= Un> 1 X*- We show that J2? is free on X, it is enough to show that PP
is isomorphic with ^f(B), where В is an alphabet with a bijection h i-> xb,
В -> X. For this, it is enough to show that: (i) X generates У: (ii) for each
nonzero Lie polynomial f(b)heB e P(B), one has f(xh)heB ± 0.
(i) Let P e LP with deg(P) = n. Then P is in En, hence for some scalars
ax, one has
Q = P - E xxxeE'n.
xeXn
Thus Q is in the subalgebra generated by En.x, hence by induction in the
subalgebra generated by X. This shows that P is in <X>, hence X generates
<P.
(ii) Arguing by contradiction, suppose that /(P1;..., Pq) = 0, for some
nonzero Lie polynomial f(bx,.. .,bq)e ^f(B) and some P1;..., Pq e X with
deg(Pj) < • • • < deg(Pg). A fortiori, there exists a nonzero polynomial f in
KfBy such that /(P1;..., Pq) = 0. Take such a polynomial with least degree,
and write it as
f = E bi9i-
i = 1
By minimality, some	, Pq) is nonzero. Since
0 = /(Л,,..,/>,) = ^ЛК.	(2-2.1)
i
we deduce that P1;..., Pq are dependent.
By Theorem 2.3 some polynomial Pi is Lie-dependent on P1;... ,P(_ P This
may be written: P{ plus a linear combination of those P,, k < i, of the same
degree as P{ = a Lie expression of the others (which are of degree less than
P{) plus an element of EB_X, with n = deg(/<). This implies that the
polynomials in Xn are not linearly independent mod E’n, which is a
contradiction.	□
2.3 AUTOMORPHISMS
If this section the alphabets are finite. Given two alphabets A, B, a
(concatenation) algebra homomorphism f: K(A> -> K(By is completely
specified by the image f(d) of the letters a in A; this is because K<A) is the
free associative algebra on A. Similarly, a Lie algebra homomorphism
(p: &(A) -> У'(В) is defined by its effect on the letters, because ^P(A) is the
free Lie algebra on A. Note that a Lie algebra homomorphism J^(A) -> У'(В)
uniquely extends to an algebra homomorphism f: К (A) —►	and that
an algebra homomorphism f: K<A) —► К (By is such an extension if and
only if /(А) с У’(В).
46	2 Algebraic properties
Recall that in Section 1.4 we have defined a linear mapping К (A) ->
К<Л>, P i—> a~ гР, for each letter a in A, by
a~1P= (P,aw)w.	(2.3.1)
we A*
Given an algebra homomorphism f: K(A) -> K<B>, we define its jacobian
matrix to be the following В x A matrix over K<B>:
J(n = (b-1f(a))beB,aeAeK<ByBx\
The terminology stems from the analogy with the commutative case. As in
the latter case, the chain rule holds. We denote as usual by J9 the matrix
obtained by applying homomorphism g to each entry of the matrix J. A
homomorphism g is called proper if g(a) has no constant term, for any letter a.
Proposition 2.6 Let А, В, C be three alphabets and f: K<A> -> K.(By,
g: К<B) -> K<C> be algebra homomorphisms with g proper. Then
J(g°f) = J(g)J(f)9-	(2.3.2)
In proof, we need the following identities:
P = (P, 1)+ E a(a*IP),	(2.3.3)
ae A
for any polynomial P in K<A>, and
a~1(PQ) = {a-xP)Q + (P, l)(a-1Q),	(2.3.4)
for any polynomials P, Q in K<A> and any letter a in A. Recall that (P, 1)
is the constant term of P. Identities (2.3.3) and (2.3.4) are easily verified when
P, Q are words, and then extended by linearity ((2.3.4) is the left-right dual
of (2.1.4)). Note that a lP = 0 when P is constant.
Proof Let a e A, c e A. Then
c \g°f){a) = с ^(/(a))) = c 1 b (/(a), I) + E bib Via))
\ L	beB
by (2.3.4)
= c 1 (/(a), i) + E gib)gib 7W
_	beB	_
= E	+ (g(b)’ i)c-1^_1/(«))]
beB
by (2.3.4)
= E c гд(ь)д(ь Via)),
beB
2.3 Automorphisms
because g is proper. Equality of the extreme members means that the
(c, a)-entry of the matrix 3{g ° /) is equal to the (c, a)-entry in the product
of 3(g) by J(f)9. This proves (2.3.2).	□
Let V denote the vector space £оеЛ Ka. A Lie algebra automorphism
(p: &(A) ->	(A) is called elementary if either qj \ Lis a linear automorphism
of V, or if for some letter a, (p(a) = a + P, where P is in <f(A\a), and (p(b) = b
for any letter b^a.
Note that, in the second case, (p1 is defined by (p~1(a) = a — P,
<p~ 1(b) = b for b + a; hence, if q> is an elementary automorphism, so is <p~ L
We call jacobian matrix of a Lie algebra endomorphism of JP(A) the
jacobian matrix of its unique extension to an algebra endomorphism of
K<A>.
Theorem 2.7 Let <p: У2(Л) -> =^(Л) be a Lie algebra endomorphism. The
following conditions are eguivalent:
(i)	(p is an automorphism',
(ii)	(p is surjective',
(iii)	the jacobian matrix of (p is right invertible in К<Л)ЛхЛ;
(iv)	(p is a product of elementary automorphisms.
The equivalence of (i) and (iv) is due to Cohn (1964).
Proof We tacitly use the following fact: if a square matrix over К<Л> is
right or left invertible, then no column or row in this matrix is 0; this may
be seen by taking the image of this matrix in К [Л j.
(i)	=> (ii) is evident.
(ii)	=> (iii) because, since (p is surjective, it has a right inverse «Д. Indeed,
define ф for any letter a by ф(а) = P for some Lie polynomial P such that
(p(P) = a', since ^(Л) is the free Lie algebra, ф extends uniquely to a Lie
endomorphism of £P(A). Let f, g be the algebra endomorphisms of Х.'<Л>
extending tp, ф respectively. Then (p ° ф = id, hence f ° g = id. By Proposition
2.6, we obtain J(f)J(g)f = IA, the A x A identity matrix. Hence J(/) is right
invertible.
(iii)	=> (iv) by induction on d((p) = ^aeA deg(<p(a)).
Since no (p(a) is zero (otherwise the jacobian matrix J(<p) of (p is not right
invertible) and since (p(a) is a Lie element, we have deg(<p(a)) > 1. If
d(q>) < | A I, then deg <p(a) = 1 for each letter a and we may write g>(b) =
Цаел^а.ь0 for some scalars ccah. Then a-1<p(h) = ctah, which shows that
J(cp) = (txa b')a beA. By hypothesis, J(<p) is right invertible in К<Л>ЛхЛ; taking
constant terms of the polynomials involved, we see that J((p) is invertible in
KA*A. Hence, (p defines an automorphism of the vector space £аеЛ Ka, and
is elementary.
48	2 Algebraic properties
Suppose now that d((p) > |Л|. Let (Pa,b)a.beA be the right inverse of J(<p).
Then we have, for any letters a, c,
X (a~1 <рФ))Рь,с = ba c.
be A
We multiply by a on the left, we sum over all letters a, and we note that (p(b)
has no constant term (it is a Lie polynomial), so that (2.3.3) gives
E <p(b)Pb,c = c.
be A
Since d(yp) > |Л|, there is some letter b0 such that deg(<p(b0)) > 2; moreover,
there is some c such that PbOtC 0, the matrix (Pb c) being left invertible.
Hence, for this c,
deg( E <P(b)Pb'C) = deg(c) = 1 < deg(<p(h0)Pi(o>c) < sup (deg(<p(h)Pft>c)).
\ЬеЛ	/	b
This shows that the Lie polynomials <p(h) are dependent. Note that none of
them is 0. Hence, by Theorem 2.3, there exists a in A and a Lie polynomial
P(b)fteB in £f\B) (with В = Л\а) such that
deg(<p(a) - P(<p(b)beB)) < deg(<p(a)).	(2.3.5)
Define an elementary automorphism ф by
Ф(а) = a - P(b)beB, фф) = b if b * a.
Define an endomorphism a of £P(A) by a = <p ° ф. Then we have a(a) =
(р(ф(а)) = (p(a — Рф)ЬеВ) = (p(a) — P((p(b)beB) because (p is a Lie algebra
endomorphism. Moreover, if b a, then a(h) = <p(<A(h)) = (рф). Hence, by
(2.3.5), d(a) < d((p). Moreover, by (2.3.2), J (a) = /(<р)/(<Д)ф. The matrices
J(<p) and are right invertible in К<Л>лхл. Moreover, M -> AP is a
ring endomorphism of К<Л>лхЛ, so that 7(<Д)Ф is right invertible. Hence
J(a) is right invertible. By induction, we conclude that a is a product of
elementary automorphisms. Hence, so is (p.
(iv)	=> (i) is evident.	□
Corollary 2.8 If n polynomials generate the free Lie algebra £P(ar,..., a„),
then they generate it freely.
Proof Let P1;..., P„ be these n polynomials. Then (p(at) = Pt defines a Lie
endomorphism of У2( Л). It is clearly surjective, so that (p is an automorphism
by Theorem 2.7. Hence, P1;..., Pn generate &(A) freely.	□
2.4 Free sets of Lie polynomials
49
2.4	FREE SETS OF LIE POLYNOMIALS
We say that a set E of Lie polynomials is free if it generates freely a Lie
subalgebra of £F(A). Recall that the derived ideal [У2, У2] of a Lie algebra
У is the linear span of the elements [P, Q], P,Qe
Theorem 2.9 Let Ebe a subset of ^(A) and £? the Lie subalgebra generated
by E. The following conditions are eguivalent:
(i)	E is free-,
(ii)	E is linearly independent modfy2, Jzf];
(iii)	E is right K(Ay-linearly independent.
Proof If E is not free, then clearly E is right K<4>-linearly dependent.
Suppose that this is the case: then some finite subset of E is К (A >-linearly
dependent, and we may suppose that E is finite. Then E is dependent, and
by Theorem 2.3 some polynomial P in E is of the form P = Q + P', where
Q, P' are in ttf Q is in the Lie subalgebra generated by E\P, and
deg(P') < deg(P). Note that Q may be written Q = Qr + Q2, where Qr is a
linear combination of elements of E\P, and Q2 e [J*9, If P' = 0, then
P — Qi e and we deduce that E is linearly dependent modf^, JS?].
If P' У 0, then replace P in E by P': we obtain a set E' which still generates
and which is still K(A )-linearly dependent. By induction on £RgE deg(R),
we deduce that E' is linearly dependent modfJZ2, У]. Hence
+ 12кеЕ\р PrR e .У], where a, fiR are scalars, not all equal to 0. This
implies that txP — txQi + 0rR 6 [J*9, JS?]; this is a nontrivial linear
combination of E (if a 0, this is clear, and if a = 0, it is a nontrivial linear
combination of E\P), hence E is linearly dependent mod [J*9,
Suppose that E is linearly dependent mod [.У9, ^]. Then clearly E is not
free.	□
The next result is the defect theorem.
Theorem 2.10 If n Lie polynomials generate поп-freely a Lie subalgebra of
&(A), then this subalgebra may be generated by fewer than n elements.
Proof As in the previous proof, we obtain that one of these polynomials,
P say, is of the form
P = Q + P',
where Q is in the Lie subalgebra generated by the others, and deg(P') <
deg(P). If P' = 0, we are done. Otherwise, we replace P by P' and conclude
by induction.	'
50
2 Algebraic properties
Corollary 2.11 Let E be a set of homogeneous Lie polynomials; then there
exists a subset of E which is free and generates the same Lie subalgebra.
Proof If E is not free then, as in the proof of Theorem 2.9, some polynomial
P in E is of the form P = Q + P', where Q is in the Lie subalgebra generated
by E\P and deg(P') < deg(P). By homogeneity, we may assume that P' = 0,
hence we can remove P from E.	□
2.5	APPENDIX
2.5.1 Rank of a free Lie algebra
If A generates a free Lie algebra &(A), then it also generates its enveloping
algebra К<Л>. Thus A generates the algebra of commutative polynomials
К[Л], because К[Л] is the quotient of К<Л> by the relations PQ = QP.
We deduce that the cardinality of A is equal to the transcendence degree of
К[Л] over K. Thus, this cardinality depends only on £?(A); we call it the
rank of зг/(Л).
A particular case is when is a Lie subalgebra of &(A). Then it is a free
Lie algebra (Theorem 2.5) and, hence, has unique rank. By Theorem 2.9(iii)
this rank is equal to the rank of the right ideal ^K(A) as a free
K<?l)-module (each right ideal in K(A) is a free K<T4)-module, by a
theorem of Cohn (1985)). The latter rank is unique, because each invertible
matrix over К<Л> is square (Cohn 1985). This again shows the uniqueness
of rank in free Lie algebras.
2.5.2 Restricted Lie algebra of characteristic p
Let К be a field of characteristic p 0. A restricted Lie algebra of
characteristic p is a Lie algebra over K, together with a mapping ->
x H-* xlP] such that
(i)	(ax)lp] = apxfpl, for any a in K;
(ii)	(x + y)[pl = x[pl + y[pl + Ap(x, y), where Ap is the Lie polynomial of
eqn (1.6.4);
(iii)	[x[p], y] = ad(x)p(y).
If за/ is an associative algebra over K, then it has a natural structure of
restricted Lie algebra of characteristic p, with x[pl = xp ((ii) and (iii) are
satisfied by eqns (1.6.4) and (1.6.1)).
Given a restricted Lie algebra let за/ be its enveloping algebra, and
the quotient of за/ obtained by identifying xp and x[pl, for each x in Then
sfp is called the restricted enveloping algebra of If (x,)lg/ is a totally ordered
basis of ЛР, then the decreasing products <p(xfl)... <p(xln), n > 0, > • • • > i„,
2.6 Notes
51
where each x, appears at most p - 1 times and where <p is the canonical
mapping У -> л/р, form a basis of Vp (see Jacobson 1962, Theorem V. 11;
Bourbaki 1971, Exercise 6 of Section 2; Abe 1980). In particular, <£ is
naturally embedded in its restricted enveloping algebra.
With this last result, one shows that the free Lie p-algebra ^,(Л) (see
Section 1.6.3) has the desired universal property in the category of restricted
Lie algebras, and that K(A) is its restricted enveloping algebra. In particular,
if (Pi)tei is a totally ordered basis of &P(A), then the set of elements
Pft	in, 0 <	..., jn < p
is a basis of К<Л>.
Theorem 1.4 has the following version in &P(A): a polynomial P is in Ур(А)
if and only if д(Р) = P 0 1 + 1 0 P (for the proof, use the Poincare-Birkhoff-
Witt theorem and the fact that d(Ppn) = Ppn 0 1 + 1 0PP" for each Lie
polynomial). Equivalently, if A has at least two elements, ad(P) = Ad(P) and
(Л 1) = 0.
2.6 NOTES
Other proofs of Theorem 2.5 are given by Cohn (1964) and Bahturin (1987);
our proof is close to the proof of Cohn. It was shown by Kukin (19776) that
the intersection of two finitely generated subalgebras of the free Lie algebra
is again finitely generated. A similar result holds in the free Lie p-algebra
(Witt 1956; see also Kukin 1972a; Bahturin 1987, Theorem 11.8).
A kind of Schreier formula holds for subalgebras of the free Lie p-algebra:
if is a subalgebra of Ур(А), of codimension d, then the rank of <£ is equal
to 1 + ра(|Л| — 1) (Kukin 1972a; Bahturin 1987, Theorem II.9). In the case
of free Lie algebras, a subalgebra of finite codimension is in general not
necessarily finitely generated.
Proposition 2.6 has an equivalent form with Fox derivatives (Fox 1953).
The equivalence of (i), (ii), and (iii) in Theorem 2.7 is from Reutenauer (1992)
and Shpilrain (1990) (see also Shpilrain 1992 for a generalization). This result
has some similarity with the jacobian conjecture in the (commutative)
polynomial algebra (Bass et al. 1982). Note that a matrix over К<Л) is right
invertible if and only it is left invertible; this follows because К<Л> is
embeddable in a (skew) field (Cohn 1985). Equivalence of (i) and (ii) in
Theorem 2.9 is due to Kukin (1977a, 1978). The analogy between free groups
and free Lie algebras, as shown in the results on subalgebras and auto-
morphisms, is further discussed by Baumslag (1972). For other results of
algebraic nature involving the free Lie algebra, see Kukin (19726), Baumslag
and Baumslag (1971) (who show that for any n, any ascending chain of Lie
subalgebras of a free Lie algebra, that are generated by n elements, is finite),
Yunus (1984), Unlu and Ekici (1986), Zerck (1989), Bryant (1991), and
Drensky (1992).
3
Logarithms and exponentials
The various characterizations of Lie polynomials obtained in Chapter 1 are
extended to Lie series, in a straightforward way. These results become really
interesting when one considers exponentials of Lie series. This leads naturally
to the Campbell-Baker-Hausdorff formula. Section 3.2 is devoted to the
canonical projections of the free associative algebra, especially the first one,
whose image is the free Lie algebra and which is obtained by a logarithm.
Section 3.3 is devoted to the computation of the coefficients of the Hausdorff
series. A generalization of this series leads again to the canonical projection.
The computation is done using descent numbers of permutations and shuffle
algebra. Several properties of the coefficients are established: symmetries,
generating functions, and recursions. The last section presents the original
methods of Campbell, Baker, and Hausdorff, which explains the introduction
of Bernoulli numbers. A closed formula is also given for the Hausdorff
series.
3.1 LIE SERIES AND LOGARITHM
Let К be a commutative Q-algebra. We define Lie series. First, suppose that
the alphabet A is finite, and let
s= z S,
и>0
in К((Л>> be written as sum of its homogeneous components. Then S is a
Lie series if each Sn is a Lie polynomial. If A is infinite and В c A, denote
by SB the projection of 5 in	That is, SB is the image of 5 under
the algebra homomorphism defined by a i—> 0 if a e A\B, b i-> b if b e B. Then
5 e K<<A>> is a Lie series if, for each finite В c A, SB is a Lie series in
K«B>>.
The mappings b, a, b, D, and r of Section 1.3 are all homogeneous and
degree-preserving. Hence, they extend (by infinite linearity) to K«A>>.
Furthermore, ad and Ad extend naturally to mappings К<<Л>> ->
3.1 Lie series and logarithm	53
Епс1к(К<<Л>» by the formulas
ad(S)(T) = [5, Г],
4d(S)(D= £ (S,u)(T,v)Ad(u)(i,).
u, ve A*
The last sum is well defined, because it is locally finite; indeed, a given word
w appears only in finitely many polynomials Ad(u)(v). This being done, we
easily obtain (from Theorem 1.4) the following result. Again, we assume that
A has at least two elements.
Theorem 3.1 Let Sbea formal series. The following conditions are eguivalent:
(i)	5 is a Lie series;
(ii)	Ad(S) = ad(S);
(iii)	b(S) = 5 ® 1 + 1 ® S;
(iv)	5 is orthogonal to each shuffle и ш v with u, ve A + , and (S, 1) = 0;
(v)	b(S) = S ® 1 - 1 ® S;
(vi)	(S, 1) = 0 and r(S) = D(S).
Proof (a) We suppose first that A is finite. Then for each series
S= E S,
n>0
written as the sum of its homogeneous parts, Sn is a polynomial because
there are only finitely many words of a given length.
Note that the mappings d, a, d, D, and r are homogeneous and degree-
preserving; more precisely, with evident notations, one has b(S)„ = b(S„),
(S ® 1 + 1 ® S)„ = S„ ® 1 + 1 ® S„, and so on. Hence, the equivalence of
(i), (iii), (v), and (vi) follows directly from Theorem 1.4 and the previous
definition of Lie series. In order to prove the equivalence of these condi-
tions with (ii), observe that, for any word w, ad(S)(w) = ad(Sn)(w) and
Ad(S)(w) = £n Ad(S„)(w); since ad(S„)(w) and Ad(S„)(w) are both homo-
geneous polynomials of degree n + |w|, (ii) is equivalent to saying Vn,
ad(S„) = Ad(S„), which by Theorem 1.4 is equivalent to (i).
Now, by Proposition 1.8, we have
3(S) = X (S, и lli v)u ® v
u,veA*
= £ (S, иш1)и®1+ Y, (5,1шг)1®1!-(5,1ш1)1®1
ueA*	veA*
+ Y, (S,umv)u®v
u,i>eA*
= 5® 1 + 1 ® 5 - (S, 1)1 ® 1 + X <S,umv)u®v.
u.ve A +
54	3 Logarithms and exponentials
Hence, (iv) implies (iii). Finally, if (iii) holds, the above equation gives us
-(S, 1)(1 ® 1) + X (S, иши)и®и = 0.
u.veA*
Hence (iv) follows.
(b) Suppose now that the alphabet A is infinite. Let В be a finite
sub-alphabet of A. Note that all the mappings of the theorem commute with
the canonical projection К<<Л>> -> K<<B>>, S SB. Hence, the general
case follows directly from the first part of the proof.	□
Theorem 3.	1 implies in particular that if a polynomial is orthogonal to
each Lie polynomial, then it is a linear combination of the empty word and
of the polynomials и ш v, и, v e A+.
Given a formal series 5 with constant term equal to 1, we may form in
К<<Л>> its logarithm; indeed, write 5 = 1 + T where Г has zero constant
term. Then the following infinite sum makes sense and defines the logarithm
of 5:
log(S) = log(l + n = L	r.	(3.1.1)
п>1 n
Similarly, one defines for each formal series U with zero constant term its
exponential by the formula
U"
eu = exp(U)= X -г-	(ЗЛ.2)
n>o n!
As usual, one has the formulas
exp(log(S)) = S, log(exp(t/)) = U.	(3.1.3)
In the next result, we assume that A has at least two letters.
Theorem 3.	2 Let S be a series with contant term 1. The following conditions
are eguivalent:
(i)	log(S) is a Lie series;
(ii)	d(S) = S®S;
(iii)	the linear mapping К(АУ -> К which sends each word w on (S, w) is a
homomorphism from the shuffle algebra К(АУ into K;
(iv)	for any series T, Ad(S)(T) = STS ~ L
Proof By Theorem 3.1, log(S) is a Lie series if and only if
5(log(S)) = log(S) ® 1 + 1 ® log(S).
(3.1.4)
3.1 Lie series and logarithm	55
Since d is a continuous homomorphism, we have d(S) = 5(exp(log(S)) =
exp(5(log(5)). Moreover, since log(5) (x) 1 and 1 (x) log(5) commute, we have
the usual identity of the exponential function: exp(log(S) ® 1 4- 1 ® log(S)) =
exp(log(5) (x) 1) exp(l ® log(S)). Since T -> T ® 1 and T -> 1 (x) T are con-
tinuous homomorphisms, this is equal to
(exp(log(S)) ® 1)(1 ® exp(log(5)) = (S ® 1)(1 ® S) = S ® S.
Finally, taking exponentials of both sides in (3.1.4), we find that log(S) is a
Lie series if and only if d(S) = S ® S. Thus (i) and (ii) are equivalent.
Now, by Proposition 1.8, we have
5(5) = £ (S,uujv)u®v.
u, ve A*
Since
S®S = £ (5, u)(S, v)u ® v,
u.veA*
we deduce that (ii) is equivalent to
Vu, v e Л*,	(S, и ш v) = (S, u)(S, v).
But this is equivalent to (iii).
Let 5 = eu and denote by g (respectively d) the continuous linear operator
on К<<Л>> defined on any word w by g(w) = Uw (respectively d(w) = wU).
Then g and d commute with each other, and hence 5w5-1 = euwe~u =
e9 e~d(w) = e9~d(w) = ead(V\w). Moreover, Ad:	-> End(K<<4)>) is
a continuous homomorphism, so that Ad(S) = Ad(eu) = eAd(U), hence
Ad(S)(w) = eAd{U}(w).
Thus, condition (iv) is equivalent to Ad(U) = ad(U), and we apply
Theorem 3.1 (ii).	□
The set of formal series with constant term 1 is a group under multiplica-
tion: indeed, the constant term of a product is the product of the constant
terms, so this set is closed under product; moreover, if S = 1 + T with
(Г, 1) = 0 then S-1 = Xn>o (— 1)"T" also has constant term 1.
The group described previously has a remarkable subgroup.
Corollary 3.3 The set of series S with constant term 1 such that log 5 is a
Lie series is a group under multiplication.
Proof Indeed, by Theorem 3.2(ii), log 5 is a Lie series if and only if
5(S) = S® S. Now, if we also have 5(T) = T® T, then d(ST) = (ST)® (ST).
Moreover, d(S ~J) = 5(5)~1 = (5 ® 5)“1 = 5 “1 ® S"1. Thus, the corollary
follows.	□
56	3 Logarithms and exponentials
The following result is the famous Campbell-Baker-Hausdorff formula.
Here, a, b are two letters.
Corollary 3.4 The series log(e° eb) is a Lie series.
This series is called the Hausdorff series.
Proof Since log(e°) = a is a Lie series, it suffices to apply the previous result.
□
Let A = {ar,... ,am}. Let a = (a1?..., am) be a path in Rm, where each
function ctj(t) is real, defined on the segment [a,/?], is continuous and of
bounded variation. Define for each word w in A* and t e [a, b] the iterated
integral (relative to a) J' dw recursively by, J' dw = 1 if w is the empty word
and if w = then f' dw is defined by the Stieltjes integral J' dw =
ft (ft du) daXs).
Corollary 3.5 The series log(£we4. (ft dw)w) is a Lie series.
Proof By Theorem 3.2(iii), it is enough to show that for t e [a, b], the linear
mapping (pt: Й<Л> -> R, wh>ftdw is a shuffle homomorphism, that is,
<p,(w lu w') = <p,(w)<p,(w') for any words w, w'. If one of these words is empty,
this is clear. Otherwise w = uu,, w' = vaj. Then by definition of the iterated
integral, we have for any letter ak and word x:
<Pt(xak)= (ps(x) dak(s),
J a
hence, by linearity,
<Pt(Pak)= f <ps(P) dotk(s),	(3.1.5)
J a
for any polynomial P.
Now, we have from eqn (1.4.2) and by symmetry
w lu w' = (u lu va^Oj + (иа, lu г)а^.
Thus, by linearity and eqn (3.1.5),
(pt(w lu w') = (ps(u lu va^ da,(s) + <ps(uaf lu v) da/s).
J a	J a
This is equal, by induction on |w| + |w'|, to
<ps(ru7)<ps(u) da,(s) +	(ps(uai)(ps(v) da/s).
J a	J a
3.2 The canonical projections	57
Since by eqn (3.1.5), ф,(ш^) = J* (pr(u) da,(r), we have (viewing <ps(Mai) as
a function of s) dt/p^ua,)) = <ps(u) da^s); similarly d(<ps(ray)) = <ps(v) da;(s).
Hence, the previous sum of integrals is equal to
(ps(vaj) d(<ps(ucZj)) + (p^uaj d((ps(vaj)) = dfp^uajq^vaj)).
J a	J a	J a
Thus, we obtain (pt(w ш w') = (pfua^qjfvaj) = (pt(w)(pt(w'), which completes
the proof.	□
3.2 THE CANONICAL PROJECTIONS
We assume that К is a Q-algebra. Define subspaces Un of К<Л> by letting
Un be the submodule of К (Ay generated by the nth powers of Lie
polynomials. This submodule has another equivalent definition. If PY,..., Pn
are polynomials, define their symmetrized product by
(Pt,...,P.) = ( X	(3.2.1)
nl aes„
where Sn is the symmetric group of order n.
Proposition 3.6 Un is the submodule of К (Ay generated by the elements
(Pn ..., Pn), where Pt are Lie polynomials.
Proof The identity Pn = (P,..., P) (n times) proves one inclusion. The
other inclusion follows from the identity
X E P,Y.
/£{1...n)	\ie/ /
which is a consequence of the inclusion-exclusion principle.	□
We shall see that there is a direct sum decomposition
К<л>= © un,
n>0
and that the corresponding projections may be computed using the convolu-
tion product ♦ of End(K<?l» defined in Section 1.5. Note that_L70 = K,
If = JT(A). Observe that the complete tensor product л/ = K(Ay ® K(Ay
(see Section 1.5) is a graded algebra, complete with respect to the graduation.
Hence, we may define
log ( x u ® w I = x —,—I E м ®u
\иеЛ*	/ k> 1 ft \иеЛ +
58	3 Logarithms and exponentials
where A+ =	The right-hand side may be written as
(_ i )* -1
X---------- X (a^ -•шак)®(а1...ак) =
к > 1 К	u i,..., Uk e A +
X X ------------,-----X (W’U1 Ш' ' 'Ш w*)wi • • •)•
we A *	\k > 1 к	/
Observe that the sum at the right of ® is finite, and is equal to a
homogeneous polynomial which has degree |w|. Hence, the formula
log! X и ® и j = X w ® ni(w)	(3.2.2)
\иеЛ* / w
defines a linear endomorphism of K<A>. Actually,
(_ I)*-1
7ti(w)= X ------7--- E (w,Ui ш-•-ш uju! . ..uk.	(3.2.3)
k> 1 К u].........uke A +
This formula shows that ^(w) is a linear combination of words obtained
by permuting the letters of w. It is equivalent to define щ using the formula
nj = log(id) in the algebra End(K<4>) with the convolution product.
Now, define for each n > 0, an endomorphism by
я, = 1 „Г,	(3.2.4)
n!
(nth power for the convolution product), or equivalently in j/ (see Pro-
position 1.10),
1 /
X w ® 7t„(w) = - £ a® 7ti(a) ,	(3.2.5)
w	n\\„	J
which may also be written as
nn=\ conc„ ° (л^п) ° dn,	(3.2.6)
n!
by eqn (1.5.7). With these definitions, we have the following result.
Theorem 3.7 The module K<A> has the direct sum decomposition
K(Aj = ф Un.
n>0
The corresponding projections К(АУ -> U„ are the endomorphisms nn.
3.2 The canonical projections	59
The projections are called the canonical projections. The proof of Theorem
3.7 may be simplified, if one admits the first assertion, which is one version
of the Poincare-Birkhoff-Witt theorem and valid in any enveloping algebra
(see Section 0.4.3).
We first prove two lemmas.
Lemma 3.8 The image of щ is contained in £T(A).
If q>: L -> К is a Z-linear mapping between two rings, then its canonical
extension £<<Л>> -> К<<Л>> maps Lie series onto Lie series. This is a
slight extension of Lemma 1.3(ii) which we need in the following proof.
Proof Let L be the shuffle algebra К<<Л>). Then we have a canonical
isomorphism л/ ~ £<<Л>). In this isomorphism, X«® « becomes the series
Хиел* auw e	where au = и e L. The /.-linear mapping £<Л> -> L
which maps each word и onto au is a shuffle homomorphism, because au = и
and L is the shuffle algebra К<<Л>>. Thus, by Theorem 3.2(iii), its logarithm
is a Lie series: log(Xu auu) = fvv e J%(4).
Let w be a fixed word and (p: L -> К be the mapping S i—> (S, w). Then,
by the remark before the proof, X <P(fib)v is a Lie series. But by (3.2.2) this
series is precisely nfw), which is therefore a Lie polynomial.	□
Let I: К<Л> -> К (A) be the linear mapping defined by I(w) = w if
we A+, /(1) = 0.
Lemma 3.9 Let <p: К <B> -> К(Л) be a concatenation homomorphism such
that (p(b) is a Lie polynomial for any letter b in B. Then <p commutes with I *k
and nt, for any к > 0.
Proof We have = log(id) = log(l + I) = X ( — l)k~1I*k/k, so it is enough
to show that <p ° I *k = I*k°tp.
Now, dk ° <p = <p®‘ ° dk because both sides are concatenation homomor-
phisms and for any letter b: bk ° (p(b) = (p(b) ® 1 ® • • • ® 1 + 1 ®
(p(b) ©•••@1 +------1-1 ® 1 ® • • • ® (p(b) (by eqn (1.5.6), because (p(b) is a
Lie polynomial) = (p®k ° bk(b). Moreover, I ° <p = (p ° I (because (p preserves
constant terms), hence I®k ° (p®k = ip®k l®k. and conck ° (p®k = <p- conck, cp
being a concatenation homomorphism. Finally by eqn (1.5.7), I*kocp =
conck ° I®k ° 5k ° (p = cp ° I*k, by putting together the previous equalities. □
Proof of Theorem 3.7 We show that
(i)	7in restricted to Un is the identity;
(ii)	any polynomial P is equal to the sum X« > о
60
3 Logarithms and exponentials
(iii)	the image of лп is contained in l/„; and
(iv)	if к / n, then 7tk(Un) = 0.
This will imply the theorem.
(i)	We have лх = log(id) = log(l + I), hence
Я, = Z '/**,	(3.2.7)
к > 1 к
in End(K</4» with its convolution product structure. This is, by eqn (1.5.7)
equivalently written as
(-1)*-1
7li = X ----------СОПСко I®k ° dk.
k> i к
Now, if P is a Lie element then, by eqn (1.5.6),
dk(P) = Р®1®---®1 + 1®Р(8)---(х)1ч-------------h 1 ® 1 ®	® P (fc terms).
(3.2.8)
This shows that I ° dk(P) = 0, unless k= 1 (because /(1) = 0). Hence,
Щ(Р) = P.
Now, we have by (3.2.8) that 5„(P") = 5„(P)" = nlP ® P ® • - • (x) P + £,
where X is a sum of terms ® Q2 ®	® Qn with at least one Qt equal
to 1. Thus 7i^"°<5„(P") = n!P ®	® P, because 7^(1) = 0 and лх(Р) = P.
This implies by (3.2.6) that л„(Р") = (1/n!) conc„ ° л^п ° <5„(P") = Pn- Since Un
is generated by the polynomials P", (i) follows.
(ii)	We have in End(K<?l>) with the convolution structure
id = exp(log(id)) = ехр(лг) = X 1 л*п = X л„,
n>o«!	«>о
by (3.2.2), Proposition 1.10, and (3.2.4), which proves (ii).
(iii)	By Lemma 3.8, we have
Imt^) с ^(A) - Ц.
Now, we have by (3.2.5)
X w ® n„(w) = X (ui ш---ши„)® (л JmJ... л1(ип))
Since the shuffle product is commutative, we can group the terms cor-
responding to the same set of words iq,..., u„; hence, using the notation of
(3.2.1):
rc„(w) = X ---------------------r^(^l(Wi),...,7l4(u„))
ui <••<«„	N(M!,...,W„)
3.3 Coefficients of the Hausdorff series	61
where < is some total order on A*, and where N(u15..., u„) is the cardinality
of the subgroup
{ст e 5„ | (uff(1),..., ua<„}) = (u15..., u„)} of S„.
Hence, by Proposition 3.6 7t„(w) is in Un and 1т(л„) c Un.
(iv)	Suppose now that В = {a}, and that in K(B) we have nfa") = 0 if
n / 1. Let P be some Lie polynomial and let be defined by <p(a) = P. Then,
by Lemma 3.9, itfPn) =	°<p(a") = nfa”) = 0. So, it remains to treat
the case of a one-letter alphabet. Observe that in j/,
(a (g) a)n = ашп ® a” = nla” (g) an;
hence,
£ an (g) nfa") = logl £ an ® an j = log(exp(a ® a)) = a (g> a.
n > 0	\n > 0	/
So we deduce that nfa”) = 0 if n 1.
Now, let к / n. Then by (3.2.8), bk(Pn) = bk(P)n is a sum of terms of the
form P'1 (g> • • • (g> Pik, where at least one ij does not equal 1. By what we have
just seen, nfP1) = 0 unless i = 1; hence, we obtain
я,(Р") = 1сопс,оя®‘ог1(Р') = 0.
nl
This proves (iv).	□
3.3 COEFFICIENTS OF THE HAUSDORFF SERIES
Recall that the Hausdorff series is the series log(ea eft), where a and b are
two letters. We shall be slightly more general and consider an alphabet
A = {ar,..., a„} of n distinct letters. By Corollary 3.3, the series
H(a15..., an) = log(eai... ea")
is a Lie series. We may write
H = Z
(p)
where (p) is a multi-index of length n and where H{p) is the finely
homogeneous component of H of multi-degree (p) = (pn ..., p„). Recall that
is the canonical projection of Q<4> on to &(A) (see Section 3.2).
Lemma 3.10 The homogeneous component H<p} is equal to
(a?	a?"\
7li----------.
\Pi! ...p„!/
62	3 Logarithms and exponentials
Proof By eqn (3.2.3), the linear mapping ях is degree-preserving and even
finely homogenous. We extend it to	by continuity. Since is the
identity on ^(A), we have 7^(5) = 5 for each Lie series S; moreover, if к 1,
7^(5*) = 0 because пл(Рк) = 0 for each Lie polynomial and Q<4> is dense
in О«Л».
This implies that
УН-Н = пЛ У -H‘)
\Р)	1 I Д-* / ।	/
р	\Л>0 *'•	/
= nt(exp(H))
/ а1'
= л1(еа' ...еа") = 7t1 X —
\(р) Pi-
• а"п\ V	(a*1 •  • аРЛ
---- = / л1|----------
Pn'J fa XpJ-.-Pn'J
Since я! is finely homogeneous, we obtain the lemma.
□
We shall give a formula for the coefficients of the words in H. For this let
A be ordered by ar < • • • < an, and consider a word w of length p in A*. A
descent (resp. a rise) of w is an index i in {1,..., p — 1} such that the ith letter
in w is followed by a smaller (resp. a greater) letter. The descent set D(w) (resp.
rise set R(w)) of w is the set of these indices. Let d(w) = |D(w)| and
r(w) = |/?(w)|, respectively the number of descents and rises of w. Let w be
written in the form
w = off. aqf...	(3.3.1)
*1	*2	x '
where two consecutive indices are always distinct, and where all exponents
are > 1. Then
D(w) = {qx +  • • + qj, 1 < j < m - 1, > iJ+1),
R(w) = {<h + • ’ • + Qj, 1 <j < m - 1, ij < ij+l}.
Moreover, d(w) + r(w) = m — 1. For example, if w = a1a1a3a2a2a1, then
m - 4, D(w) = {3, 5}, R(w) = {2}, d(w) = 2, r(w) = 1.
Similarly, we define the number of descents and of rises of a permutation
in S„, viewed as a word of length л on	naturally ordered. Note
that if we S„, then d(w) + r(w) = n — 1.
The eulerian polynomial En(x) is the polynomial
£,(x)= X x*'1.
<reS„
For example, Ex = 1, E2 = 1 + x, E3 = 1 + 4x + x2, £4 = 1 + 1 lx +
1 lx2 + x3, Es = 1 + 26x + 66x2 + 26x3 + x4.
3.3 Coefficients of the Hausdorjf series
63
The homogeneous eulerian polynomial En(x, y) is
= /-*£, - = X x1My™
\y / aeSn
(3.3.2)
It will be convenient to put Eo = 0.
Define a linear function r. Q[x] -> Q by
s(xk) =
(-0*
к + 1 ’
In other words, we have for any P in Q[x]:
r °
J -1
s(P) =
P(x) dx.
(3.3.3)
Theorem 3.11 Let w be written as in (3.3.1), and Pi,.--,p„ its respective
lengths in the letters ax,... ,a„. Then the coefficient of w in log(efll ... e°"), or
equivalently the coefficient of w in the projection nfa^' ... a^/p^.... p„l), is
equal to
s(xd(l x x)r [J Eq.(x, x + \)/qf.),	(3.3.4)
\	1 < j < m	J
where d (respectively r) is equal to the number of descents (respectively rises)
of w.
We need some preliminary results. Given a word и = b{... bp of length p,
and a permutation о of Sp, define
ucf = ba(V)... ba(p}.
This is a right action of Sp on the words of length p: indeed, if we denote
by u(i) the ith letter of u, then we have (uo)(i) = u(oi), so that (u(oaf)(i) =
u(oa(i)) = (uo)(oti) = ((uo)a)(i), hence u(oa) = (uo)a.
If W is a set of permutations, we define its eulerian polynomial by
E„(x)= X x*”.
aeW
and its homogeneous eulerian polynomial by
E^(x. y) = £ хд',У”-
aeW
Lemma 3.12 Let w be as in (3.3.1), of respective length Pi,...,p„ in the
64	3 Logarithms and exponentials
letters аг,..., a„, and let и = a{'... a?". Let IV = {cr e Sp | uo = w}, where p
is the length of и and w. Then
E^fx) = Pl' Pn' xdtw> П M*)'
-	l<j<m
and
E^(x, y) = Pf Pd xd1w>yriw> П
q1'... qm!	i <j<m
Proof Let /15..., Im be the consecutive intervals of {1,..., p}, of respective
lengths q^... ,qm. Let G be the subgroup of S„ which leaves these intervals
invariant; hence о e G if and only if ст(/у) = f, for j = 1,..., m. Since и i—> uo
is a right action, and since G evidently fixes w, we have о e IV, aeG =>
ста e IV. Hence, W is a union of left cosets ctG. In each coset C, there is a
unique element ctc whose restriction to each interval f is increasing. We
claim that there are pf.... p„l/qf.... qm\ such cosets in W. Indeed, W is also
a single right coset modulo the subgroup fixing u; the latter has pf.... p„!
elements, hence so does W and so W is union of pt!... p„!/|G| cosets of G.
The claim then follows because G has qf... .qm\ elements.
Observe that for r = qr + • • • + qp i.e. the greatest element of Ij, and for
any ст in W, one has r e D(ct) ore D(w). Indeed r e D(ct) о ст(г) > ст(г + 1) =>
и(ст(г)) > и(ст(г +1)) (because и is an increasing word) о w(r) > w(r + 1)
(because w = ист) о w(r) > w(r + 1) (by the choice of г) о r e D(w); similarly,
ст(г) < ст(г + 1) implies w(r) < w(r + 1), hence the reverse implications also
hold.
In particular, since ctc and w have no descent in Ij\{r}, we have
D(ctc) = D(w).
Next we claim that if a e G, then Р(стса) is the disjoint union D(w) о D(a).
From the previous observation, we deduce that Л(стса) n +    + q-,
1 < j < m - 1} = D(w). Since ctc|/7 is increasing, and since a permutes f,
we have for i in /7\{qi +------h qj}: ct(i) > a(i + 1) о CTca(i) >	+ 0-
This proves the claim.
From the latter claims, we deduce that
E„.M = Eac(x)Ea(x) = xdloc>Ea(x) = хл">Еа(х),
hence E„(C(x) = xdMEc(x). Evidently, E0(x) = E„(x) ... E,„(x), so that
EM = У E„c(x) = хл” У Ec(x) = xJ<’> р‘---р"\ E „(X)... E,_(x).
For the second identity of the lemma, note that ст e Sp implies г(ст) + d(o) =
3.3 Coefficients of the Hausdorjf series	65
p — 1; thus by (3.3.2)
( y\	n I n ! /
Ew(x,y) = yp'1Ew( I = yp~l — /".( ) П /'’'E (x,y)
\y/ qf- qm' \y/	i<j<m
because r(w) = m — 1 — d(w) and p = qr +    + qm.	□
The right action of the symmetric group Sp on the words of length p
extends linearly to a right action of the group algebra Q[5p] on the linear
span of the words of length p. It is convenient to extend it to all of О<Л>
by the formula: wo = 0 if о e Sp and w is a word of length / p. If 5 is a subset
of {1,..., p — 1}, we denote by D-s the sum, in Q[5p], of the permutations
whose descent set is contained in 5. In the next lemma, we identify each
permutation in Sp with the corresponding word on {1,... ,p}. Recall that
the convolution product * has been defined in Section 1.5.
Lemma 3.13 Let p1,...,pk (k > 1) be positive integers of sum p and
S = {pi,Pi + p2,---,Pi +  ' + Pk-i} the corresponding subset {1,...,p — 1}.
Factorize the word 1 2 ... p as ur... uk with ImJ = ph for each i.
(i) One has
DsS = 0(ui ш‘ ‘ ’ш ukf
where в is the linear involution of Q[5p] sending each permutation onto its
inverse.
(ii) Let qn denote the linear endomorphism of such that qn(£i Pi) = Pn,
for each polynomial P = £ Pt written as the sum of its homogeneous compo-
nents. Then, for each polynomial of degree p, one has
pdsS = (%>* *qPk)(P)-
The right action of Q[5p] defined previously leaves the canonical scalar
product on invariant (this scalar product has A* as orthonormal basis;
see Section 1.1). This implies that for any words u, v of length p and any
permutation о in Sp, one has (u, vo) = (uo~ l, v). Thus, if x is in Q[Sp], we
obtain
(u, vx) = (ив(х), v),	(3.3.5)
which means that the adjoint of the linear endomorphism of Q<4>: v i-> vx
is и i—> u6(x).
Proof (i) A permutation о appears in the sum Dt S if and only if its descent
3 Logarithms and exponentials
66
set is contained in S, that is
a(i) > o(i + 1) => i e S,
for any i in {1,..., p — 1}. Moreover, a permutation a appears in lu • •  lu uk
if and only if for any i in {1,..., p — 1 }\5, the digit i + 1 appears at the
right of i in the word a(l)... a(p), that is
i ф S => a - 1(i) < a - 71 + 1),
or equivalently
a-1(i) > a-1(i + 1) => ie S.
This implies the lemma, since the sums D-s and lu- • -lu uk+ j are
multiplicity-free.
(ii) The adjoint of the linear mapping P^PD-S is by (i) and (3.3.5)
the linear mapping P i—> P(u1 ш   -lu uk). The adjoint of the mapping
qPi* -  *qPk is the mapping qpi *'• • -*'qPk, where *' is the convolution
product defined in Section 1.5; indeed, the adjoint of cone and d are
respectively 3' and sh (Proposition 1.9), so that the adjoint of f*g =
conc(/ ®g)cd is sh ° (/♦ ® g*)° д’, where /♦, g* denote the adjoint of
f, g; hence, the above assertion is implied by the fact that qn is self-adjoint.
Thus, it is enough to show that for any word w, one has
w(u j lu    lu uk) = (qpi qPk)(w).	(3.3.6)
Suppose that |w| = p. Let w = vx ... vk be the factorization such that |rf| = p,.
Then, by definition of the right action of Sp, one has
w(ul LU • • - LU Uk) = LU • • - LU Vk.
On the other hand, by eqn (1.5.8), we have
(qPi *' ’ -*'qPk)(w) = shk°(qpi®- •  ® qPk) ° d'k(w)
= shk°(qpi®-  ®qPk)( X Wi®--®wk]
=	^P,(W1) ® • •  ® qPk(wk)^
= shk(Vi ® - ®vk)
= Vi LU-  -LU Vk.
If |w| / p, then the same calculation shows that both sides of (3.3.6) vanish. □
Lemma 3.14 Let I: K(A) —►	be the linear mapping which sends each
3.3 Coefficients of the Hausdorff series	67
nonempty word onto itself and the empty word on 0. Then for any к > 1 and
any word w of length p, one has
I*k(w)= J wD^s,
|S| = k-l
where the summation is over subsets S of {1,..., p — 1}. Moreover
S<={1...р-I} |dI + 1
Proof Because of (3.2.7), we have only to verify the first equality. We have
I = Хл > i Яш with the notations of Lemma 3.13(ii). Thus
^k= £
pi,•.•, Pk > i
Since qPi * • • • ♦ qPk = cone ° (qPi ® • • • ® qpk) ° d sends w onto 0 if рг + • • •
+ pk / p, we deduce that
I*k(w) = X (<?₽> *• ’ -*^)(w).
pi + • • • +pk =p
Since (p15..., pk) и-► {pi, pi + p2,..., Pi + • • • + pk_ J is a bijection from
the set of sequences of length к of positive integers of sum p onto the set of
subsets of cardinality к — 1 of {1,..., p}, we obtain the first equality from
Lemma 3.13(ii).	□
Proof of Theorem 3.11 The equivalence in the statement follows from
Lemma 3.10. Let x = a[' ... a*”lpt\ ... pn\. We have only to show that
(л i(x), w) is equal to the last formula of the theorem.
By Lemma 3.14, we have
(-l)|s|
(n1(x),w)= X ------------------~(xD^s,w),
S<=(1...p- 1} |o| + 1
where p is the common length of и = nf1 ... a$n and w. By definition of DsS,
this is equal to
( — nisi	( — nisi
s |S| + 1 ~sp	s |S| + 1
D(a)^S	ua = w
with h = \/pYl.. .pn'.. This is equal to
ua = w D(a) S S |S| +1
Since each subset D(o) is contained in (p~1 [d{a>) subsets of {1,..., p — 1} of
68	3 Logarithms and exponentials
cardinality d(o) + i, and since i may be equal to 0,..., p — 1 — d(o), we
obtain
(7t!(x), w) = h X
U(T= w
= hsl X
\U<7 = W
p-l-d«7) (_	+ i Л, _ 1 _
i=o d(o) + i + 1 \ i
i = 0	\ i /	/
= hsl X xd(<T)(l
\U<7=W
= hsl X + x)r(ff)
\U<7 = W	,
= hs(Ew(x, x + 1)),
where IV = {a e Sp | uo = w}. Hence the theorem follows from Lemma 3.12.
□
Let m, qu..., qm, d, r be non-negative integers. We denote by
A(d, r;qY,..., qm) the expression (3.3.4). By Theorem 3.11, it is the coefficient
of w, written as in (3.3.1), in the series log(efll ... efl"), if d (respectively r) is the
number of descents (respectively of rises) of w; note that in this case
d + r = m — 1.
Corollary 3.15 The coefficient k(d, r; qY,..., qm) does not depend on the order
of the sequence ql,...,qm. Moreover, if d,r are interchanged, then it is
multiplied by (— I)41 + ’ ’ ’
Proof The first assertion is an immediate consequence of the definition of
z. For the second, observe first that En(x, y) = En(y, x); indeed, the mapping
which sends each digit i onto n + 1 — i is a bijection S„ -> Sn (where each
permutation is viewed as word on {!,...,«}) which interchanges descents
and rises. Moreover, by (3.3.2), we have En( — x, —y) = (—i)n~1En(x,y),
because d(o) + r(o) = n — 1 for any о in S„. Now, using (3.3.4) and interpret-
ing the linear form s as an integral (see (3.3.3)), one performs the change of
variables x = — 1 — y, and the result follows by the previous observations.
An alternative proof (in the case where d + r = m — 1) is to apply to
log(efll ... efl") the anti-automorphism a of Q<<4>> which sends each Lie
series S onto — S (Lemma 1.7).	□
The previous corollary shows that if w is a word of even length with an
equal number of descents and rises, then its coefficient in log(eai ... ea") is
0. This is the case in particular for the coefficient of a word of even length
beginning and ending with the same letter in the Hausdorff series log(ea eb).
3.3 Coefficients of the Hausdorff series	69
Observe that by definition (3.3.4) of A, and the fact that Er = 1, we have
A(d,	qm) = Jfd, r; 1,..., 1, <h, • • •, qm)
(any number of ones). We write X(d, r) for X(d, r; 1,..., 1).
Corollary 3.16 The coefficient X(d, r) is equal to (— i)ddlrl/(d + r + 1)!. In
other words, if each permutation in S„ is considered as a word on A =
one has in 0<Л>
n1(12...n) = £ ---------- x <*•
aeSn П \ d{O) )
Proof By definition of A and the fact that Efx, y) = 1, we have
f 0
A(d, r) = s(xd(l + x)r) = xd(l + x)rdx.
J -1
Note that A(d, 0) = x xd dx = (— V)d/d + 1. Hence the result is true for
r = 0. Now, let r > 1. Then
f 0
X(d, r) = xd(l + x)rdx
J -1
’1	T	f° r
= ------x‘, + 1(l+x)r	— I -------xd + X(1 + x)r"1 dx,
|_d+l	J-i	J-i^+1
by integration by parts. Hence, by induction on r, we have
X(d,r)=-----r—X(d+ l,r- 1)
d + 1
r (-\)d+1(d+ l)’(r — 1)!
“ ~d+\ (d + r + 1)!
(- l)dd!r!
“ (7+7+1)! ’
The last assertion of the lemma follows from the fact that 7^(12... n) is a
linear combination of permutations in Sn, by (3.2.3).
Corollary 3.17 Let d, r, m be nonnegative integers with d + r = m — 1. The
generating function in Q[[f 15..., tm]] of the numbers 2.(d, r; qY,..., qm) is
70	3 Logarithms and exponentials
In order to understand why the right-hand side is a formal power series in
t15..., tm, we make the following observations: if a formal power series is an
alternating function of t15..., tm, then its quotient by A =	— rj) *s
still a formal power series; now, the right-hand side is symmetric in t15..., tm,
hence its product by A is alternating; furthermore, the quotient
(Г; — tj)/(efi — e'J) is a formal power series, hence this product too.
We begin with a classical lemma. Recall that we have put £0 = 0. In
particular, л vanishes if some qt = 0.
Lemma 3.18 The exponential generating function of the eulerian polynomials
is
tn i	i)
„>i nl ец — x
Proof Let us write En(x) = a(n, k)xk, where for n > 1, a(n, k) is therefore
the number of permutations in Sn with к descents. Each permutation in S„ +15
when viewed as a word, is obtained by shuffling a permutation in S„ with
the digit n + 1. A closer look then shows that
a(n + 1, к + 1) = (к + 2)a(n, к + 1) + (n — k)a(n, k).
We have, moreover, the initial conditions a(n, 0) = 1 if n > 1 and a(Q, k) = 0
for к > 0. On the other hand, we write
1 _ pMx- 1)	fn
S = ----------= У b(n, k)xk —.
e,(x — x	„Л	и!
Then it is seen (by putting t = 0, then x = 0) that the same initial conditions
hold for the b(n, k). Moreover, a simple computation shows that
S + 1 + (x — x2) — + (tx — I) — = 0.
dx	dt
By taking the coefficient of xk+1tn/nl, we obtain
b(n, к + 1) + (к + l)b(n, к + 1) — kb(n, к) + nb(n, к) — b(n 4- 1, к + 1) = 0.
Hence the b(n, k) satisfy the same recursion as the a(n, k). We deduce that
a(n, k) = b(n, k), hence £ En(x)tn/nl = S.	□
From the lemma, we easily deduce the exponential generating function of
the homogeneous eulerian polynomials:
fl- eHx-y) t(y-x) _ j
У En(x, y)- =----------------=---------------.	(3.3.7)
n > i nl у e‘(x y) — x у — x et(y x)
71
3.3 Coefficients of the Hausdorjf series
Indeed, we have by (3.3.2) and Lemma 3.18
tn	i x\ tn
X En(x,y)~ = X
i>1	n'.	„>i \yj n'.
= 1 V F (A (^)"
Уп>1 \y) n'.
| 1 _ Qyt[(xly)~ 1]
” у еУЫхМ-и _ x/y
| _ gt(x-y) ^Цу-х) _ |
У qUx-у) _ x у — x et(y~x>
We deduce from (3.3.7) the formula
tn e' — 1
X En(x, x + 1) =----------------------= F(x, t).
n! 1- x(e - 1)
(3.3.8)
Proof of Corollary 3.17 We have by Theorem 3.11
X z(d, r; , qm)tr ... C
qi....qm
= X /xd(l+x)r П EqfX,X + 1)/<7j4rT •••*£”
41..qm \	1 < j<m	/
= s(xd(l+x)r X П Eq.(x,x + lyt^/qf)
\	41,...,4m 1 < j<rn	j
where we have extended the linear function s: Q[x] -> Q to s:
QM((ri> • • • Л™)) Q, coefficient-wise. This is by (3.3.8)
s(xd(l + x)r П X Eqfx, x + Ow)
/	Q(J - |	\
= si xd(l + x)r П ----------------- .
\ iZUl -x(e''- 1)/
Now, we have, for some elements Nj in the field of fractions of Q((t b ..., tm)):
л	т-г e,J - 1	™	Nj
xd(l + x)r П ------------------= У----------------,
1 - x(etJ - 1) J=1 1 -	- 1)
because the left-hand side is a rational fraction in x with degree of numerator
equal to d + r = m — 1, which is smaller than the degree of the denominator,
equal to m, and because the latter is a polynomial in x with simple roots.
72	3 Logarithms and exponentials
To compute Nh multiply both sides by 1 — x(en — 1) and put
x = \/(eh — 1), hence 1 + x = e,7(e,i — 1):
(e'1 - 1)V
because d + r = m — 1.
Now, observe that for T in Q[x]((t!,..., tm)), s(T) = — h(T)\x= _15 where
h is the Q((rls..., tm))-linear operator of О[х]((П,..., tm)) which maps xk
onto xk + 1/(k + 1). Hence, h is an integration operator, and so
M s
_____N<_
1 — x(e"
\ m — N-
Thus, the generating series of the z’s is
In order to justify completely the previous computations, the reader may
find it useful to perform them in L ®QWf)) Q[x]((t)), with Q((0) =
Q((tb ..., tm)) and L the field of fractions of the latter.	□
Recall that the Bernoulli numbers Bn are defined by the exponential generating
function
У и _ X
„Го т cx - 1 ’
(3.3.9)
It is easily verified that Bo = 1, Вг = —1/2 and that x/(ex — 1) + x/2 is an
even function, which shows that Bn = 0 for odd n > 3. The first few values
are B2 = 1/6, B4 = -1/30, B6 = 1/42.
Corollary 3.19 For p,q > \,the number A(0, 1; p, q), i.e. the coefficient of apbq
in log(ea eb), is equal to
(~1)P f <f\R
plql Л1Ш P+^k'
Proof By Corollary 3.17 and (3.3.9), we have
3.3 Coefficients of the Hausdorjf series
73
ey _ 1
X Ж1; i, j)xlyJ = x ex ----------
j> i	e — ey
ex — 1
ey — C
ey — e'
ey — e:
ex(y — x)(ey — 1)
ey — ex
e^x-l
= “}' + (£ ХУW ~
\i>0 t'	/
= -y + (X X	1)
\i>0 0<j<i \J/	I-	/
= -y+I z I	^‘+<
i>0 j<ik> 1 V/	l- K-
By change of variable p = i — j, q = к +j (hence к < q, j = q — к and
i = p + q — k), this sum becomes
cP
-У+ X X , „„m».-»-
P>O,<J>1 p'-q'. i <k<q(q — ky.k'
Thus, we have for p,q> 1:
Л(о,1;А,) = Ц^ f	□
p'q' k = i\kj
Let us call size of an и-tuple (qY,..., <?m) of positive integers the number
(t/i — !) + ••• + (qm — 1). If this size is zero, then the qt are all equal to 1,
and the coefficient A(d,	is given by Corollary 3.16. The next
results give recursion relations between these coefficients, allowing us to
compute them by induction on the size.
Corollary 3.20 The following relations hold, for non-negative integers d, r,
Яи-чЯт with qm> 1:
(i) i(d + l,r;qlt... ,qm)= X ~ #>Ж П <h, • • •, qm-1, qm + 1 - if
0 <i <,qm
(-1)'
(ii) X(d, r + 1; qY,..., qm) = X ——Bjfd, r; q^,..., qm-p qm + 1 - i).
74	3 Logarithms and exponentials
Proof The first lines of the proof of Corollary 3.17 give us
X Ж r; qY,..., qjtf .. .t4” = st xd(\ + x)'
...q™	X
etj — 1
1 — x(eh — 1),
П
< i<
(3.3.10)
Now, we have
etm — 1
1 — x(e,m — 1)
so that
etm — 1
1
etm — 1
1 — x(e'm — 1) etm — 1
1 — x(etm — 1)
Since s is Q((tm ))-linear, we deduce
et] — 1
1
1 — x(e° — 1)/ etm — 1
= si xd(l + x)r
etj — 1	\
1 - x(e° - 1)/
etj — 1
:m 1 - x(e° - 1).
Note that the first series of the right-hand side does not involve the variable
tm. Now, take on both sides the coefficient of t]1... t*'”, using that by (3.3.9)
l/(e‘m — 1) = Xr>o (Br/r!)t^71. On the left-hand side this coefficient is, by
(3.3.10),
X ~A(d,r;q1,...,qm_1,qm + 1 - i).
0 < i < qm I-
On the right-hand side, it is A(d + 1, r; qr,..., qm) by (3.3.10).
This proves the first equality. To prove the second, use Corollary 3.15.
□
Corollary 3.21 (same hypothesis as in Corollary 3.20). The following relations
hold:
(0 (Qm + 0Ж C qY,..., qm- 15 qm + 1)
= -dt(d, r + \;qY,...,qm)~ rA(d + 1, r; qt,..., qm)
X t(d + l,r + 1; <?i, •-	---, <7m);
1 < к < m — 1
qfc + «fc' = 4k
3.3 Coefficients of the Hausdorjf series	75
(ii) + W Г 4i, • • •, qm-1, qm + 1)
= /.(d + 1, r; Qi,..., qm) + Ж r + 1; qu ..., qm)
+ E i(d + 1, r +	q'm,q'^).
q'm +q'm=qm
Proof A straightforward computation shows that F(x, t), given by (3.3.8),
satisfies dF/dt = (x + (x + 1))F + x(x + 1)F2 + 1. Hence, we deduce by
(3.3.10) and because s is Q[[tm]]-linear:
~ ( E Ж C qY,..., qJW ... t4”]
\<Ii./
= / (V-^1 + -v)r П ж g-)Y)
\ \	l<j<m J J
= sfxd+1(l + x)r fl F(x, tj) j + sfxd(l + x)r+1 fl F(x, tj)\
+ s[xd + 1(l + x)r+1 П F(x, tj)F(x, tm)2 )
+ s(xd(l + x)r П F(X, tj)Y
\	1 < j < m - 1	/
By (3.3.8), the third term is equal to
s(xdtI(l + x)'+1 I ( П (?'EJx,x+I)/,/)
X t^^E^X + 1)E^(X’X +
Hence, by taking the coefficient of t^' ... t’"1 in the above equality, we obtain
the second relation of the corollary.
Since s(/(x)) = fLi /(x) dx, we have
s( ~ (*d+ ‘(1 + *)'+ ‘F(x, F(X, („))) = 0.
\(7X	/
Now, we have
^=Fix,,r,
dx
which implies, similarly to what has been done above,
(d + 1)Ж r + 1; qY,..., qm) + (r + 1)Ж + 1, r; qt,..., qm)
+ X E A(d+ l,r+l;qi,...,qfq'j,...,qm) = O.
1 < j < m q’j + q’j = qj
16	3 Logarithms and exponentials
The first equality of the corollary follow from this and the second one. □
3.4 DERIVATION AND EXPONENTIATION
We begin by a general formula on the derivative of an exponential in a
noncommutative algebra. We concentrate only on the algebraic aspect and
omit the convergence conditions; in our applications all infinite sums are, as
usual, locally finite.
So, let M be an associative algebra over some commutative ring К
containing Q. Let cp be an algebra homomorphism M -» M. A((p, cp)-
derivation D is a function M -> Al, linear over K, and such that
Vx, у e Al, D(xy) = D(x)(p(y) + <p(x)D(y).	(3.4.1)
We note as before by ad(a) the operator: b h-► [a, b].
Theorem 3.22 Let f(t) = Xn > о antn be a formal series with coefficients in K.
With the previous definitions, one has for x in M
D(f(x))= X ^ad((px)k~\Dx)f(k\<px).
к > 1 к-
Observe that when Al is commutative and D a derivation (i.e. cp = id), then
one obtains the classical formula
D(/(x)) = D(x)/'(x).
Proof The formula is К-linear in f, so it is enough to prove it when
f(t) = tn. Let us write (px = a, Dx = b. Then the right-hand side is
X — ad(a)k~ 1(b)n(n — 1) ... (n — к + l)an~k,
k>i k!
which is equal to
" (n\
X , \ad{a)k-\bffin-k.	(3.4.2)
к = 1 \kj
We show that (3.4.2) is equal to D(x"), by induction on n. The case n = 0 is
clear; let us assume equality for n. Then, by (3.4.1) and induction
n (и\
D(x"+1) = D(xx") = ban + aD(xn) = ban + X ]a ad(a)k~1(b)an~k.
k = l \kj
3.4 Derivation and exponentiation
77
Because au = [a, u] + ua = ad(a)(u) + ua, this is equal to
ban + \'\ad(a)k(b)an k + J \]ad(a)k \b)a‘
k = i\kj	k = i\kj
I n \
= ban + У I )ad(a)k~1(b)an+1~k
k=2 \k ~ 1/
" I n\
+ X ( , }ad(a)k~1(b)a‘
к = 1 \kj
= ba" + £
k = 2
n \ n
к — 1/	\ к
ad(a)k 1(b)a'
n)ad(a)n(b) + \]ban
nJ	\1/
и + 1 /n +
= E ( , \ad(a)k 1(b)al
к = 1 \ к J
□
Corollary 3.23 (same hypothesis)
D(ex) =
ead(<px) _ |\
----------- (Dx) e*x.
ad(tpx) J
Proof In the theorem, let f(x) = ex. Then we obtain
D(ex) = У — ad((px)k~ Y(Dx) е9Х
k> i kl
which proves the corollary.	□
Consider again the Hausdorff series H defined by
e° eb = eH
in Q<<n, b>>. Let
H = I Hn,
n>0
where Hn is the part of H which is homogeneous of degree n with respect to
a. In particular, Ho = b. Recall that the Bernoulli numbers Bn are defined by
(3.3.9).
Corollary 3.24 One has
HY = f= a + b] + У ~ad(b)2n(a).
1 \eadW- 1J	n>i(2n)l
78	3 Logarithms and exponentials
Proof Let (p: Q<<n, b>> -» Q<<n, b>>, the homomorphism defined by
<p(a) = 0, (p(b) = b. Let D be the linear endomorphism such that
jw if |w|a = 1,
D(w) = <
[0 otherwise,
for any word w. Thus D keeps only the words which have one a. Then
Hr = D(H). Observe that for any words и and v, one has
D(uv) = (p(u)D(v) + D(u)(p(v).
Hence, D is a (<p, <p)-derivation of Q<<n, b>>. By Corollary 3.23 we thus
obtain
/ead(<pH) _ l\
a eb = D(ea eb) = D(eH) = -------------- (DH) е”я.
\ ad(<pH) J
Since (pH = Ho = b and DH = we deduce
/pad(b) _ i\
o= ----------- (Я,),
\ ad(b) /
which implies
H' = (e^,t, _ , )W-
The second equality is a consequence of the definition (3.3.9) of the Bernoulli
numbers.	□
If D is a derivation of the algebra M (as above), then the linear mapping
exp(D): M -> M given by
exp(D)(x) = X “
и>0 И!
is an algebra homomorphism M -> M. This is a consequence of the
well-known Leibnitz formula
в”(х>’)= E ("Wu
In the case of Q<<n, b>>, we denote by S(d/db) the derivation of Q<<n, b>>
which maps a on to 0, and b on to S (where S e Q<<n, b>>).
Corollary 3.25 The Hausdorff series is equal to
( d \
H = exp Hr — (h).
\ dbJ
3.4 Derivation and exponentiation
79
In particular,
1 / а у
Hn = Hi — (b).
nl V dbj
Observe that this result shows again that H is a Lie series. Indeed, we have
Hn + 1 = l/(n + 1)(HX a/ah)(H„), so it is enough to verify that H1 d/db maps
Lie series into Lie series. But 0 = 1^ d/db is a derivation of Q«a, b>>, so
its restriction to 3\a, b) is a derivation of Lie algebra, i.e.
D(IP, QI) = LDP, Q] + [P, DQ],
and the generators a and b are mapped onto 0 and which is a Lie series
by Corollary 3.24. Thus Lie series are mapped into Lie series.
Proof Let D = H^d/db) and p = exp(D). Then D is a derivation and p a
homomorphism. We have by Corollary 3.23 (with (p = id) and Corollary 3.24
D(eb) =
~ad(b) _ < \	/ ad(b) _
-------)(Db)eb =--------------
ad(b) )	\ ad(b)
(Hx)eb
ead(b) _ j
u</(b)
ad(b)
Qad(b) _ |
(a) eb = a eb.
Thus, by induction, because Da = 0, we obtain
Dn(eb) = an eb.
Hence,
p(eb) = X 1 Dn(eb) = X an eb = ea eb.
n>0 til	n>Otll
Since p is a continuous homomorphism, we have
p(eh) = eM<b),
which implies eM<b) = e" eb. Hence, p(b) = H, as required.
All the previous results have a dual version. Indeed, we have with the
notations of Corollary 3.23:
/1
D(ex) = e4
_ Q~ad(<px)'
ad(<px) ,
(Dx).
Let us write
h= £ h;
л>0
80	3 Logarithms and exponentials
where H'n is the part of H which is homogeneous of degree n in the letter b.
Then
= (—= ^ + Ka’ M + У ad(a)2n(b),
1	_ e-ad(a)J^ 2L „^o (2м)!	7
and
( d\
H = exp Hj — (a).
\	da J
Finally,
H’ =
1
n!
( , d\n
- (a).
\	da J
3.5	APPENDIX
3.5.1 Friedrichs’ criterion generalized
Theorem 3.1 implies that if P is a polynomial with constant term 0, then P
is a Lie polynomial if and only if P is orthogonal to each proper shuffle
и ш v (и, v e A +). The following more general result (essentially from Cohn
(1954)) holds: let и be a positive integer and denote by the linear span
of products PY ... Pq, where each Pi is a Lie polynomial and q < n. Then a
polynomial P is in =5? if and only if P is orthogonal to each shuffle products
ux ш - • ш мл+1, with u,e A + . This may be proved by using the Poincare-
Birkhoff-Witt theorem (Theorem 0.2) and the generalized coproduct 3n + x
of Section 1.4. It is also a consequence of Theorem 5.3(iii).
3.5.2 Chen series
Let a = (an..., am) be a path in IR"1, as in the hypothesis of Corollary 3.5.
We call the Chen series of a the formal series (fa dw)w; by Corollary
3.5, the logarithm of this series is a Lie series. If /?: [b, c] -> is another
such path, whose initial point is equal to the endpoint of a, then the Chen
series of the path obtained by concatenating a and f is equal to the
concatenation product of the Chen series of a and f.
If m = 2 and a is the horizontal path from (0,0) to (0, 1) and ft the vertical
path from (0, 1) to (1, 1), then their Chen series are respectively eai and e°2;
hence, the path obtained by concatenating them has Chen series eai e°2; hence
the latter is the exponential of a Lie series by Corollary 3.5. This shows that
this result extends the formula of Campbell-Baker-Hausdorff (for these
results see Chen (1957)).
3.5 Appendix	81
3.5.3 Free group
Let F(A) denote the free group over A. Then the mapping а не" (a e A)
extends uniquely to an injective homomorphism from F(A) into the group
of formal power series in Q<<4>> with constant term 1. Each element of
the image is of the form es, for some Lie series S (Corollary 3.3); see Magnus
et al. (1976, Section 5.10) for results on this image. Actually, each mapping
of the form a 1 + a + a2S(a) extends to an injective homomorphism from
F(A) into	(see Bourbaki 1972, Theorem II.5.1); for example, the
Magnus transformation (see Section 6.3).
3.5.4 Dynkin’s formula
The following formula for the Hausdorff series H is obtained by an
application of Theorem 1.4(v) (which states that for each homogeneous Lie
polynomial P of degree n, r(P) = nP, where r is the right-to-left bracketing
mapping), and an easy computation:
H = \og(eaeb) = X (^-1)* = X —X —J
fc>i к	k>i к \P+q>o Р-q'/
= X v----------------------ap'bqi ... aPkb4k
к pM- • • • pM-
where the summation is over all к > 1,	   , pk, qr,... ,qk>0 with
p,r + q,; > 0 for i = 1,..., k. Hence, one obtains
H = ^^-_____________________1______________________1________
к Pi + • ' • + Pk + qY + • • • + qk Pi'- •   pk'-qi'- - - - qk'-
x r(ap'bq' ... aPkbqk)
(Dynkin 1947, 1949, 1950).
3.5.5 Zassenhaus formula
There exists a unique sequence (Рл)„>2 of homogeneous Lie polynomials in
a, b such that Pn is of degree n and that
e°+b = e° eb ePi ePi...
Let us write S = P + 0(4") if S — P has no terms of degree < n. The main
lemma to prove the previous formula is to show that if S = 1 + 0(4") is the
exponential of a Lie series, than S = epT, where P is a homogeneous Lie
polynomial of degree n and T = 1 + 0(4" + 1) is the exponential of a Lie
series; use Corollaries 3.3 and 3.4, and the fact that log(eaeb) =
a + b + 0(42).
82	3 Logarithms and exponentials
A similar product formula may be proved by applying the homomorphism
(p ® id: .e/ -»	to the formula of Corollary 5.6 where (p is any shuffle
homomorphism О<Л> -» Q, e.g. cp defined by ea+b = (p(w)w (cf. Theorem
3.2(iii)). See also Fer (1958).
3.6	NOTES
Condition (iii) in Theorem 3.2 is due to Ree (1958), and condition (iv) to
Cartier (1956), who also proved Corollary 3.3. Corollary 3.4 is the formula
due to Campbell (1897, 1898), Baker (1905), and Hausdorff (1906). Corollary
3.5 is from Chen (1957). The formula in the proof of Proposition 3.6 is taken
from Higman (1956). The second part of Theorem 3.7 is from Solomon
(1968b) and Reutenauer (1986b), while the first part is an equivalent version
of the Poincare-Birkhoff-Witt theorem; see Section 0.4.3 or Dixmier (1974,
2.4.6). A formula similar to eqn (3.2.3) may be found in Hain (1986). Lemma
3.8 is from Ree (1958). Joyal (1986) shows that the analytic functor ‘tensor
algebra’ is the exponential of the analytic functor ‘free Lie algebra’.
Lemma 3.10 is taken from Bourbaki (1972, II.6.4 Remarque 1). Theorem
3.11 is due to Goldberg (1956). For its proof, we have followed Helmstetter
(1989), who proved Lemma 3.12. Lemma 3.13(i) is from Garsia and Remmel
(1985), and Lemma 3.14 is from Garsia (1990). Corollaries 3.15, 3.17, and
3.19 are due to Goldberg (1956). Corollary 3.16 is from Solomon (1968b).
The recursion formulas of Corollaries 3.20 and 3.21 are due to Helmstetter
(1989). Another proof of Theorem 3.11 was given by von Waldenfels (1966a):
he considers more general series of the form 0(/i(«i). • •/(«„) — 1), where the
ft are one-variable power series with constant term 1, and g is any series in
one variable. The fact that the computation of the Hausdorff series may be
an application of a computation in the algebra of Sn was noted by Dynkin
(1949). For more on eulerian polynomials, see Foata and Schiitzenberger
(1970).
In Section 3.4, we have followed more or less the original proofs of Baker
(1905) and Hausdorff (1906). Theorem 3.22, in particular, is due to Baker
(1905, p. 340), and Corollary 3.23 is essentially already in Campbell (1897);
this result explains why the Bernoulli numbers step in. Corollary 3.24 is in
all of the three papers (Campbell 1898; Baker 1905; Hausdorff 1906). Finally,
the formula of Corollary 3.25 is from Baker and Hausdorff. For more on
the Campbell Baker- Hausdorff formula, including a history of it, see Czyz
(1992). For results on exponentials of derivations of the free associative
algebra, see Lenormand (1969/70).
We have not discussed convergence of the Hausdorff series. For this, the
reader may see Dynkin (1950), Cartier (1956), Bourbaki (1972, Chapter II,
Sections 7 and 8), Lazard (1963), Michel (1973/74, 1974, 1976), and
Postnikov (1986, Lecture 6).
3.6 Notes
83
Several numerical computations have been made for the coefficients of the
Hausdorff series: Poetsch and von Waldenfels (1964), Michel (1973/74, 1974,
1976), Maltey (1988), and Koseleff (1991).
The Campbell Baker Hausdorff formula has been applied to the Burnside
problem for groups: Magnus (1950) and Michel (1976). A continuous
Campbell Baker Hausdorff formula may be found in Magnus (1954),
Bialinicki-Birula et al. (1969), and Michel (1974).
4
Hall bases
Hall sets of trees are introduced in Section 4.1. A rewriting system on
standard sequences of Hall trees is defined and shown to be confluent,
invertible, and always terminating. As a consequence, each word has a unique
decreasing factorization into foliages of Hall trees. By interpreting the nodes
of a Hall tree as the Lie bracket, one obtains a basis of the free Lie algebra,
and the corresponding Poincare- Birkhoff-Witt basis, in Section 4.2. In the
last section it is shown that this basis construction gives exactly the same
bases as those obtained by elimination in Chapter 0. In the appendix it is
shown amongst other things that Hall bases are in some sense optimal.
4.1 HALL TREES AND WORDS
Recall that the free magma M(A) has been defined in Section 0.2. Each tree
t of order at least 2 will be written
t = (t', t"),
where t' (respectively t") is its immediate left (respectively right) subtree. There
is a canonical mapping from M(A) onto the free monoid A*, defined by
f(a) = a if a is in A, and f(t) = f(t')f(t"), if t = (f, t") is of degree >2. Note
that |t| = |/(t)|. We call f(t) the foliage of t.
Let H be a subset of M(A). Then H is called a Hall set if the following
conditions hold:
• H has a total order <;
• A is contained in H;
• For any tree h = (h', h") in H\A, one has h" e H and
h < h".	(4.1.1)
• For any tree h = (h', h") in M(A)\A, one has h e H if and only if
h’,h"eH and h'ch”,	(4.1.2)
and
either h'eA, or h' = (x, y) and у > h";	(4.1.3)
4.1 Hall trees and words
85
Observe that each subtree of a Hall tree is again a Hall tree. If a Hall set
is fixed, we call Hall tree an element of this set.
Proposition 4.1 Hall sets exist.
Proof Let < be a total order of M(A) such that for any tree t of
degree >2, t = (t', t"), one has
t < t".
Such a total order surely exists. Define recursively a set H by A £ H and,
if h = (h',h”), then heH if and only conditions (4.1.2) and (4.1.3) are
satisfied. Then it is clear that H is a Hall set.	□
Actually, there are continuously many Hall sets, for each finite A with
|Л| > 2; this is left as an exercise. The reader may also verify that each Hall set
is obtained as in the proposition.
Example 4.2 The alphabet is A = {a,b}. In Fig. 4.1 we give the trees of
degree < 5 of a Hall set in M(A), ordered by < from left to right and line
after line.
Fix a Hall set H in M(A). We define a standard sequence of Hall trees to
be a sequence
s = (/ip ...,/i„),
(4.1.4)
with n > 1, hi e H and for any i, either hi is a letter, or hi = (/i-, h”) with

(4.1.5)
a b
Fig. 4.1
86
4 Hall bases
In other words, a sequence of Hall trees is standard if for each tree h in it,
which is not a letter, its immediate right subtree h" is greater or equal to
every tree located at the right of h in the sequence. Observe that a sequence
of letters is always standard; moreover, each decreasing sequence of Hall trees
(i.e. hr > • • • > hn) is standard; indeed, if /i, = (h'h h"\ then by (4.1.1) we have
h" > h(, hence h” > hi + l,..., hn.
We call rise of a sequence (4.1.4), an index i such that ht < hi + 1; by
abuse, we also say that the rise is the couple (hhhi + l). We call inversion
of (4.1.4) a couple (i,j) such that i < j and ht < hj. Note the opposite
inequality, contrary to the usual inversions; this is because our ‘ideal’ Hall
sequences are decreasing. Thus, a Hall sequence is decreasing if and only if
it has no rise, or equivalently, no inversion. A legal rise of (4.1.4) is a rise i
such that
^i+ 1	+	> К-	(4.1.6)
We now define a rewriting system on the set of standard sequences. Let s in
(4.1.4) be standard, and i a legal rise of s. Then we write
s s',
where
s' = (/ip ..., hi_1, (hi, hi+1), hi + 2,..., hn).	(4.1.7)
In other words, s' is obtained by composing, in M(A), the two trees which
form the legal rise in s. Observe that s' is standard: indeed, we have either
ht;e A, or hi= (h'i, h'-) and then	by (4.1.5), which shows that
(hi,hi+l) is a Hall tree, by (4.1.2) and (4.1.3); moreover, (4.1.6) holds, and
for j ~ 1,..., i — 1, either hj is a letter, or h'- > hi+l (by (4.1.5), because s is
standard) > (/i,, hi+l), by (4.1.1). Hence s' is standard.
We denote by the reflexive and transitive closure of the binary rela-
tion
Theorem 4.3 (i) For any standard sequences s, s15 s2 with s st and s1* s2,
there exists a standard sequence t such that s{ t and s2 t.
(ii)	For each standard sequence s, there exists a sequence of letters t such
that t s.
(iii)	For each standard sequence s, there exists a decreasing standard
sequence t such that s t.
Property (i) is expressed by saying that is confluent. Property (ii) means
that is invertible.
Proof (i) Write s t if t is obtained from s by applying the binary
relation p times. Then
We have s s{, s s2, and we may suppose p, q 0 and 4- s2.
4.1 Hall trees and words
87
Fig. 4.2
Suppose first that p = q = 1. Let s be as in (4.1.4), Sj as in (4.1.7) and
s2 = (Лр ..., hj_ p (hj, hj+1),..., hn) where j is a legal rise of the sequence
s. We may suppose i < j. We claim that i + 1 <j: suppose indeed that
i + 1 = j; then hi+ j = hj, hence hi+l < hj+ p because j is a rise; on the other
hand, i is a legal rise of 5, so that by (4.1.6), hi+i > hi+2 = hj+a
contradiction, and the claim is proved.
We show that (hj, hJ+1) is a legal rise of s, and (ht, hi+l) is a legal rise of
s2. Indeed, the first assertion is clear, because what is at the right of hj+1 in
Sj is the same as what is at the right of hj+ j in s, and because (hj, hj+ J is
a legal rise of s. The second assertion follows from the fact that (hh hi+ J is
a legal rise in s, and that hi+1 > hj+l (by (4.1.6), because j + 1 > i + 1) >
(hj, hj+1), by (4.1.1) because (hj, hj+ J is a Hall tree.
Define now
t = (hp..., h,_ p (ht, hi+1), hi+2, ...,hj_!, (hj, hj+ J, hj+2,..., hn).
Then we have st -> t and s2 -> t.
Suppose now that p = 1 and q > 2. Then we have for some s2: s -> s'2,
s'2——► s2. By the previous part of the proof, we have Sj -> t', s'2 -> t' for
some t'. By induction on q, we find t such that t’ t, s2*+ t. Thus Sj t,
which concludes this case (see Fig. 4.2).
We now treat the general case s sx, s s2. We may find s', with s -> s\,
s\ ——► Sp By the previous case, we find t' such that s\ t', s2 t'. By
induction on p, we find t such that s^t, t' t. Hence, s2 t, which
concludes the proof (see Fig. 4.3).
(ii)	Let s be as in (4.1.4). We argue by induction on £"= j |/i, | — n. If this
number is 0, i.e. s is a sequence of letters, we are done. Otherwise take i such
that hi is not a letter, and for any j = 1,..., i — 1, either hj is a letter, or
88
4 Hall bases
h] > h". Such an i exists: take the left-most ht which is not a letter. Then let
t' = (h15..., /i,_ 15 h'h h", hi+l,..., hn).
The sequence t' is standard: indeed, either /1- is a letter, or (Л'-)" > h" (by
(4.1.3), because (Л-, h") is a Hall tree) > hi + 1,... ,hn (by (4.1.5), because s is
standard); moreover, either h" is a letter, or (h")" > h" (by (4.1.1)) >
hi +15..., h„; finally, for j = I,... ,i — 1, either hj is a letter or h'j > h" (by
the choice of i) > h\, by (4.1.2). This shows that t' is standard. Furthermore,
(h'b hf) is a legal rise of t', because If < h" by (4.1.2) and h" > hi+1,... ,hn,
by (4.1.5), s being standard. Hence, t’ -> s. By induction, we find a sequence
of letters t such that t t'. Hence, t s.
(ii	i) If s is not decreasing, then s has at least one rise. Let i be the right-most
rise. Then
ht < hi+1 >hi+l>---> h„,
with s as in (4.1.4). This shows that this rise is legal, hence we find s' with
s -> s'. By induction on the length of the sequence, we find a decreasing
sequence t such that s' t. Hence, s t.	□
Corollary 4.4 Each word win A* has a unique factorization
w = f(hf)... f(h„), H,hi >   >hn.
Before proving this, we call foliage of the sequence s in (4.1.4) the word
f(s) = f(hf>... f(hn).
Observe that by (4.1.7)
s —» t implies f(s) = f(t).
(4.1.8)
4.2 Hall and Poincare-Birkhoff-Witt bases
89
Proof Let w = ar ... ap, for some letters a,, and consider the standard
sequence s = (<ц,..., ap). By Theorem 4.3(iii), there exists a decreasing
sequence t = (7ц,..., hn) of Hall trees such that s t. By (4.1.8), we deduce
that w admits the desired factorization.
Suppose now that w has two factorizations as above: w = /(h1).../(h„) =
/(Ц)... f(tp). Consider the standard sequences s = (7ц,..., hn), t —
(Ц,..., tp). By Theorem 4.3(ii), we find sequences s', t' of letters such that
s' s,t' t. By (4.1.8), we have f(s') = f(s) = w = f(t) = f (t'), hence s' = t'.
By Theorem 4.3(i), we conclude that s u, t и for some sequence u. Since
s, t have no rises, this is only possible if s = t.	□
We call Hall word the foliage of a Hall tree.
Corollary 4.5 Each Hall word is the foliage of a unique Hall tree.
Thus, we may identify Hall trees and words. We call Hall set in A* the
image under f of a Hall set in M(A), with the corresponding total order.
Example 4.6 The Hall words corresponding to the trees of Example 4.2 are
a2ba, a2ba2, a2bab, a2b, a, a2b2, a2b2, abab2, ab, ab2, a2b2, ab2, b.
Corollary 4.7 Each word has a unique decreasing factorization into Hall
words.
If 71 is a Hall word, let t be the unique Hall tree such that h = f(t). If h is
not a letter, then t — (t', t"\. let h' = f(t'), 7i" = f(t"\, then h = h'h”, and we
call this factorization of h its standard factorization. For later reference, we
note several inequalities on Hall words, which are immediate consequences
of the definitions and of (4.1.1), (4.1.2), and (4.1.3): if h is a Hall word with
standard factorization h = h'h", then
h'<h”,	(4.1.9)
and
h<h".	(4.1.10)
Now, let к be another Hall word with h < k. Then hk is a Hall word with
standard factorization hk if and only if
either h e A, or h" > k.	(4.1.11)
4.2 HALL AND POINCARE BIRKHOFF WITT BASES
Let К be a commutative ring with unit and consider a fixed Hall set. We
define for each Hall word h a Lie polynomial Ph in &K(A): if a is a letter,
90
4 Hall bases
then Pa = a; if h is a Hall word of length >2 with standard factorization
h = h'h", define Pb = [P'b, P*]. It is clear by induction that each Ph is an
homogeneous Lie polynomial of degree equal to the length of h; furthermore,
Ph has same partial degree with respect to each letter as h.
Example 4.8 The Lie polynomials of the Hall set of Examples 4.2 and 4.6
are
Pfl2fcfl = [[a, [a, b]], a] = 3a2ba — 3aba2 + ba2 — a2b,
Paiba2 = EEEfl, Ea, a], a] — 6a2ba2 — 4aba2 + ba4 — 4a2ba + a4b,
Pa2bab = [[a, [a, 6]], [a, 6]] = a2bab — 3aba2b + 2ba2b — a2b2a
+ 4ababa — 3ba2ba — ab2a2 + baba2,
?а2ъ = Ea, b]] = fl2b ” 2aba + ba2,
Pa = a,
Ра3ь2 = Ea- Efl, EEa, b], b]]] = a2b2 — 2a2bab + Aababa — ab2a2 — a2b2a
— 2baba2 + b2a2,
Pa2b* = Efl, EEEfl, ^3, b], b]] = a2b2 — 3abab2 + 3ab2ab — 2ab3a + 3bab2a
— 3b2aba + b2a2,
Pabab2 = EEfl, b], EEfl, b], 6]] = abab2 — 3ab2ab + 2ab2a — ba2b2
+ Ababab — 3bab2a — b2a2b + b2abab,
Pab = Efl, b] = ab - ba,
Pab3 = EEEfl, b], b], b] = ab2 — 3bab2 + 3b2ab — b2a,
Pab. = EEEEfl, b], b], b], b] = ab4 - 4bab3 + 6b2ab2 - 4b3«b + b4a,
Pa2b2 = Efl, EEfl, b], b]J — a2b2 — 2abab + 2baba — b2a2,
Pab2 = EEfl, b], b] = ab2 — 2bab + b2a,
Pb = b.
The previous example shows a fact which is clear from the definition: in
order to compute Ph, one has simply to interpret in the tree t corresponding
to h each node as a Lie bracketing. We call the polynomials Pb the Hall
polynomials. These polynomials form a basis of the free Lie algebra, as the
next result indicates. Recall that the free associative algebra K.(A') is a free
К-module having the canonical basis A*.
Theorem 4.9 (i) The Hall polynomials form a basis of the free Lie algebra
(viewed as a K-module).
4.2 Hall and Poincare-Birkhoff-Witt bases	91
(ii) The decreasing products of Hall polynomials
Phi...Phn, l^e H, hx>    >hn	(4.2.1)
form a basis of the free associative algebra (viewed as a K-module).
The second part of the theorem is the Poincare-Birkhoff-Witt theorem
applied to the basis (Ph)heH of 2T(A). We shall not use the latter theorem
here and actually prove (ii) first and then deduce (i). The proof is constructive
and allows us (i) to express each Lie polynomial in the basis of Hall
polynomials, without computing these, and (ii) to express each polynomial
in the basis of decreasing product of Hall polynomials, again without
computing these products.
Consider again a standard sequence of Hall trees (or words)
s = (h1,...,hn),	(4.2.2)
with a legal rise i. We define
2,(s) = (Zip • • •, h^ 15 /i./il + 1, /il + 2, • • •, hn),	(4.2.3)
which is a standard sequence (see (4.1.7)); we also define
Pi(s) = (/i15..., hi_ 15 /il + 1, hi, hi+2,h„),	(4.2.4)
obtained by interchanging /j, and /il + 1 in s. Observe that p,(s) is a standard
sequence: indeed, either hi+1 is a letter, or /i"+1 > hi+1 (by (4.1.1)) > h;,
because i is a rise. Observe also that p,(s) has one inversion less than s.
Let s be a standard sequence. Define a derivation tree T(s) of s to be a
labelled rooted tree with the following properties: if s is decreasing, then T(s)
is reduced to its root, labelled s; if not, T(s) is the tree with root labelled s,
with left and right immediate subtrees T(s') and T(s"), where s' = ^(s),
s" = p;(s) for some legal rise i of s (e.g. the right-most rise, cf. proof of
Theorem 4.3(iii)) and where T(s'), T(s") are derivation trees of s', s"
respectively. Observe that T(s) always exists, and is finite.
Example 4.10 With the Hall set of Example 4.2, a derivation tree associated
with s = (a, b, b, b) is shown in Fig. 4.4.
Observe that in a derivation tree, the leaves are labelled by decreasing
sequences. For s as in (4.2.2), define P(s) = Phl ... Phn.
Lemma 4.11 For each standard sequence s, P(s) is the sum of all P(t), for t
a leaf in a fixed derivation tree of s.
Example 4.12 From Example 4.10 and Lemma 4.11 we deduce
abbb = Pabbb + 3PbPabb + 3P2bPab + P2bPa.
92
4 Hall bases
Fig. 4.4
Proof The lemma is a consequence of the definitions (4.2.3) and (4.2.4) of
2i(s) and pt(s), of that of T(s) and P(s), and of the identity in K<A>:
PhPhl+t = LP^Phl+3 + Phl+iPhi = Phih. + 1 + Pht+1Ph.- □
Proof of Theorem 4.9 If w = аг ... a„ (a, e A), then s = (at,..., an) is a
standard sequence, hence w = P(s) is, by Lemma 4.11, a sum of polynomials
of the form (4.2.1).
We have now to show that the polynomials (4.2.1) are linearly indepen-
dent. For this, we may assume that the alphabet is finite. Note that, in this
case, the К-module of homogeneous polynomials of degree d admits the
basis Ad. Since Ph is a homogeneous polynomial of degree |Л|, Corollary 4.7
gives a canonical bijection between the polynomials (4.2.1) which are of
degree d and the words of length d. Hence, these polynomials form a basis:
indeed, if M is a free К-module of rank n, then each family of n elements
which generates M is a basis of M.
A particular case of (ii) is that the Hall polynomials are linearly indepen-
dent. Hence, it remains to show that £P(A) is linearly generated by the Hall
polynomials. We may suppose that the alphabet A is finite. Since each letter
is a Hall polynomial, it suffices to prove the following claim (here and in
the sequel, h = h'h" denotes the standard factorization of a Hall word /1).
For any two Hall polynomials Ph, Pk, their Lie bracket [Ph, Pfc] is a
linear combination over Z of Hall polynomials P( with |/| = \h\ + \k\ and
/" < sup(/i, k).
This will be shown by induction on the couple (|h| + |k|, sup(/i, k)), where
these couples are lexicographically ordered: (d, k) < (d15 kJ if either d < dr
or d = di and к < kP As there are only a finite number of Hall words of
bounded length, the induction is correct. We may suppose that h < k, because
[Ph, Pk] = — [Pk, Ph], and [Ph, Ph] = 0. If h is a letter, or if h = h'h" with
h" > k, then by (4.1.11), hk is a Hall word and hk is its standard factorization.
Thus [Ph, Pk] = Phk and we have (hk)" = к < sup(/i, k), because h < k. So we
4.2 Hall and Poincare-Birkhoff-Witt bases	93
may assume that h = h'h" and h" < k. By (4.1.10), we have also h < h”, hence
h<h" <k.	(4.2.5)
Using the Jacobi identity
[[Л Q], К] = [[P, /?], Q] + [P, [Q, /?]],
we obtain
ЕЛ, Pkl = [IA, P„J, Pkl = [ЕЛ-, Pkl, Ph I + LPh , [Рл , PJ]•
Since |h'| + |fc| and \h" | + |fc| are both strictly less than |/i| + |fc|, the induction
hypothesis implies that
[ft. ft] = Z «,Pu„	[ft--. ft] = Z №
‘	j
for some integers a,-, and Hall words ut, Vj such that:
= l^'l + I&I, lul = \h"I + I&I, u" sup(h', k) and v'- < sup(/i", k).
Note that by (4.1.9) h' < h", hence by (4.2.5), the two previous inequalities
imply:
tff<k, v';<k.	(4.2.6)
Thus we obtain
[ft, ft] = Z «<[ft,, ft ] + Z Mft-, ft,]-	(4-2.7)
‘	j
We have |u,| + |h"| = lh'| + |fc| + |/f'| = |h| + |/c|, and sup(u,, h") < к =
sup(/i, k) by (4.1.10), (4.2.6) and (4.2.5). Hence [PUi, Ph.J is by induction a
Z-linear combination of Hall polynomials P( with /" < sup(u,, h") < sup(/i, k),
and |/| = |uj + |/i"| = |h| + \k\. Similarly, we have |h'| + |i?;| = |/f| + |h"| +
|fc| = |/i| + |fc| and sup(/i', Vj) < к = sup(/j, k) because of (4.1.9), (4.2.5), (4.1.10)
and (4.2.6). By induction we deduce that [Ph,, PVj] is a Z-linear combination
of Hall polynomials Pt with |/| = |h'| + |гу| = |h| + |fc| and I" < sup(/i', Vj) <
sup(/i, k). With (4.2.7), the previous discussion proves the claim. □
It is worthwhile to write down explicitly the algorithm underlying the
proof of Theorem 4.9(i). For this, let us denote by Pt the Lie polynomial
obtained by interpreting in the tree t each node as a Lie bracketing; formally,
Pa = a if a e A, and Pt = [Pt, Pr] if t = (tr, t"). The algorithm takes as input
a linear combination of trees J att and gives as output a linear combination
of Hall trees £ [ihh such that £ atPt = £ PhPh.
Consider the linear combination J att. If all the trees involved are Hall
94
4 Hall bases
Fig. 4.5
(4.2.10)
trees, then we are done. Hence, take some tree t which is not a Hall tree,
and consider a subtree s = (s', s") of t which is not a Hall tree and such that
s’, s" are Hall trees (each leaf is in A, so is a Hall tree, hence s exists). If
s' > s", then
replace s by (s", s') in t and at by — at.	(4.2.8)
If s' = s" then
remove t from the linear combination.	(4.2.9)
So, we may assume that s' < s". If s' is in A, or s' = (x, y) with у > s", then
s is a Hall tree, which was excluded. Thus у < s", and
replace t by the sum of the two trees and t2 obtained by
replacing s = ((x, y), s") respectively by ((x, s"), y) and (x, (y, s")).
Then go back to the beginning. The fact that this algorithm stops and gives
the desired output is implicit in the proof of Theorem 4.9(i). We illustrate
the algorithm by an example.
Example 4.13 We take the Hall set described in Example 4.2. The input
linear combination is the tree of Fig. 4.5. After step (4.2.8) applied to the
subtree whose root is circled, we obtain the output given in Fig. 4.6 (the
coefficient is indicated at the root). After step (4.2.8) again, we obtain the
output given in Fig. 4.7. Now, step (4.2.10) gives the linear combination of
Fig. 4.8. Finally, step (4.2.8) again gives the final linear combination of Hall
trees (these are not indicated in Example 4.2, but are Hall trees by definition
of the latter); see Fig. 4.9. As another example, take as input the tree in Fig.
4.10. After step (4.2.10), we obtain the linear combination in Fig. 4.11. Step
(4.2.9) removes the first tree, and the output is the remaining one, which is
a Hall tree.
4.2 Hall and Poincare-Birkhoff Witt bases
95
Fig. 4.6
Fig. 4.8
A closer look at the proof of Theorem 4.9 shows that the Lie algebra of
Lie polynomials is the free Lie algebra, independently of the Poincare-
Birkhoff-Witt theorem and of the proofs in Chapter 0. Indeed, it shows that
each relation between the elements Pt (t in the free magma on A) may be
deduced from the defining relations of Lie algebras: distributivity of the
bracket, antisymmetry, and Jacobi identity.
96
4 Hall bases
Fig. 4.9
Corollary 4.14 Let A be finite with q elements. The number of Hall words of
length n, and the dimension of the space of homogeneous Lie polynomials of
degree n are equal to
1E MQn'd
U d\n
where p is the Mobius function.
4.2 Hall and Poincare-Birkhoff-Witt bases	97
Proof Call a„ the number of Hall words of length n. By Corollary 4.7, we
have the following identity of generating series
Indeed, Corollary 4.7 implies that one has in
1	X«<= A a weA* h 1 h
where the product is taken over all Hall words in decreasing order; eqn
(4.2.11) follows by applying the homomorphism 2<<Л>> -> Z[[t]J which
sends each letter onto t. Take the logarithmic derivative of (4.2.11) and
multiply by t:
qt „ kak tk
C^qt ~	l~t*'
Hence,
X ^" = X X kaktki= X f"(x^aA
n> 1	k> 1 i> 1	n> 1 \d|n /
Thus q” = Xd|n dad, which implies by Mobius inversion
nan = X p(d)qnld.
d | n
The fact that an is equal to the dimension of the space of homogeneous Lie
polynomials of degree и is a consequence of Theorem 4.9(i).	□
In the course of the proof of Theorem 4.9, we have also obtained the
following result.
Corollary 4.15 Each word, when written in the basis (4.2.1), has coefficients
in ГЧ1.
Corollary 4.16 If L is a subring of K, then ^K(A) n L(A) = J?L(A).
Proof Let Ре^(Л)п£<Л>. Then by Theorem 4.9(ii) P is a linear
combination over L of polynomials of the form (4.2.1). Since P is a Lie
polynomial over K, Theorem 4.9(i) implies that only those polynomials
(4.2.1) with n = 1 may appear in this linear combination. Hence, P is in
&l(A). The reverse inclusion is clear.	□
98	4 Hall bases
Corollary 4.17 If К is a ring without torsion over %., then Theorem 1.4 is
valid for K.
Proof Indeed, К is a subring of the Q-algebra К Q and we apply
Theorem 1.4 and Corollary 4.16.	□
4.3 HALL SETS AND LAZARD SETS
The following result shows that the bases of £f(A) constructed in Sections
0.3 and 4.2 are the same. The proofs are very technical.
Theorem 4.18 (i) Each Hall set is a Lazard set.
(ii) Each Lazard set is a Hall set.
For trees s, t in M(A) and n > 0, recall the notation (st") for the tree
(.. .(s, t),..., t), with n ts.
We begin by a lemma.
Lemma 4.19 Let H be a Hall set in М(Л) and suppose that A has a greatest
element c. Let X = {(acn) | a e Л\с, n > 0}. Then H' — H n M(X) is a Hall
set in M(X), where M(X) is identified with a submagma of М(Л). Moreover,
H = {c} u H', and c is the greatest element of H.
Proof Observe that by (4.1.1) each tree in H is smaller than or equal to its
right-most leaf. Since с = тах(Л), we deduce that c is the greatest element
of H. Observe also that the submagma of Af(A) generated by X is a free
magma, isomorphic with M(X).
Let heH \X, with h = (hf, h"), h',h"eM(X). Then by (4.1.2),
h', h" e H n M(X) = H' and h' < h"; moreover, if h' is not in X, then
ti = (x, y) with у > h” by (4.1.3).
Conversely, let h', h" e H' with h' < h" and either /f e X or h' = (x, y) with
x, yeM(X) and у > h". Then by (4.1.2) and (4.1.3), h = (Л', h") is in
H n M(X) = H', except perhaps in the case where h' eX\A. But in this case,
we have h' = (x, y) with у = c, hence certainly у > h", because c is the greatest
element of H. This shows that H' is a Hall set in M(X), because (4.1.1) is
evidently satisfied.
Let t eH\c. We show that t e M(X). If |t|	= 1, then	t e Л\с	x. If	|t| > 2,
t = (t', t") with t', t" e H and t' < t" by (4.1.2); hence	t' / c. Suppose	t" Ф c.
then t', t" are in H\c, hence in M(X) by induction, hence t e M(X). Suppose
t" = c; we show by induction on |t'| that	teX: if	[f| = 1,	then t'	e A\c,
hence (t', t") = (f, с) e X; if |t'| > 2, then	(r')" > t"	= c by	(4.1.3),	which
implies (t')" = c and by induction t'eX; this implies t = (f, с) e X, by
(4.3.3)
4.3 Hall sets and Lazard sets	99
definition of X. Thus, finally, H\c £ M(X) n H = H'. The reverse inclusion
is clear.	□
Let t0,..., tn be elements of M(A). Define subsets To,..., Tn+1 of M(A)
recursively by To = A. and
Tk+ i = {ttD \p>0,te Tk\tk} for к = 0,..., n. (4.3.1)
Observe that Tk\tk £ Tk+l. It is immediate to deduce that for i > 1,
Tk\{tk,...,tk + i^}^Tk + i.	(4.3.2)
Observe also that if x e Tk+1, then either x = (rtf) as in (4.3.1), with p > 1,
or x e Tk. Hence
x e Tk+1\T0 => xeTi+1 for some i < k,
x = (ttf) for some p > 1 and t e TfL-
Suppose now, that for к = 0,..., n, one has tk e Tk. Then
he Tk+1,..., Tn+ j.	(4.3.4)
This is actually a consequence of Corollary 0.8. A different proof is the
following: observe that 7^ + 1 £ M(Tk), the submagma generated by Tk.
Hence, Tk+1 £ for к > 0. Now, each tree in T, either has t0 as a proper
subtree, or is in Л\Г0, so a fortiori in M(A\t0); thus, the same holds for each
tree in M(Ti), which implies t0 ф hence t0 ф Tk+1, for any к > 0. This
proves (4.3.4) for к = 0. For к > 1, one proceeds inductively, by noticing first
that M(7]) is canonically isomorphic to the free magma generated by Tr, and
similarly for each M(Tk).
Suppose now that £ is a Lazard set and E a closed subset of M(A) such
that eqns (0.3.3), (0.3.4), and (0.3.5) hold. Then we have
Tk n E £ L for к = 0,..., n.	(4.3.5)
Indeed, if t e Tk n E and if t ф L, then t Ф t0,..., tn by (0.3.4), so that teTn+1
by (4.3.2). But this contradicts (0.3.5).
Proof of Theorem 4.18 (i) Let E be a finite, nonempty and closed subset of
M(A). We show by induction on |E| that for each Hall set H, H satisfies
conditions (0.3.3}-(0.3.5) defining a Lazard set (with H in place of L). Denote
by A' the finite alphabet of letters which actually appear in E. Let
с = шах(Л') and X = {(acn) | a e A'\c, n > 0}. Let H be a Hall set in M(A).
Observe that H n M(A') is a Hall set in M(A'). Then by Lemma 4.19 applied
to M(A') and its Hall set H n M(A'), H' = H n M(X) is a Hall set in M(X),
where M(X) is identified to a submagma of M(A).
100
4 Hall bases
The set £' = E n M(X) is a finite closed subset of M(X), of smaller
cardinality than E, because ceE\M(X). If E' = 0, then each letter b
appearing in E is not in M(X), hence equal to c: this shows that E involves
only the letter c, and E n H = {c}; since c e A and X n E = 0, conditions
(O.3.3)-(O.3.5) defining a Lazard set are satisfied in this case.
If E' / 0, then by induction on |£|, we conclude that for some n > 1,
H' n £' = {tj > • • • > t„}, with tj e X = T\, tt e £• = J | p > 0,
te \t, _ j} for i = 2,..., n, and Г'+ j n £' = 0. Let t0 = c e A, To = A
and for i = 1, . . . , и 4- 1,7] = {tt/L J | p > 0, t e 7]_ Дг,- Д. Then Т\ £ 7] for
i = 1,..., n, and an easy induction on i shows that if t e 7Д 7Д then t involves
some letter in A\A'. Hence, we have Tn+1 n E = T'n+1 n £ (because
£ £ M(A')) = 0. Since c is the greatest element in A', it is also the greatest
element in £ n 77, by (4.1.1). If t e £ n H and t / c, then t e M(A'), and by
Lemma 4.19 applied to M(A'), we deduce t e H'; in particular, t e M(X),
hence t e H' n £' and t = t, for some i = 1,..., n. Thus EnH —
{t0 >    > tn}, which completes the proof of (i).
(ii) Let £ be a Lazard set in M(A). We show that it is a Hall set.
Suppose that h e L\A. Then h = tk +15 for some closed subset £ of M(A)
and with the notations of (0.3.3), (0.3.4), and (0.3.5). Since he Tk+1by (0.3.3)
we deduce by (4.3.3) that for some i < k, h = (ttP) with p > 1. Hence by
(0.3.4) h" = ti > tk + 1 = h and (4.1.1) holds.
Clearly L contains A, because for any letter a, L n {a} is nonempty by
(0.3.4). Now let h = (h', h") be in L. We show that (4.1.2) and (4.1.3) hold.
Choose a closed finite subset £ of M(A) containing h and take the notations
of (0.3.3), (0.3.4), and (0.3.5). By hypothesis, we have h = tk+ x e Tk+ x for some
k, and by (4.3.3), we have h = (rtf), p > 1, t e T^. Thus h' = (ttP~J), h" = t,.
Since h’ e Ti+l n £, we deduce from (4.3.5) that h' e L, hence that h’ = tj for
some j. By (4.3.4), we must have i 4- 1 < j 4- 1, hence by (0.3.4) h" =
t; > tj = h'. Suppose that h' ф A. Then either p > 2, and (h')" = tt > t, = h",
or p = 1, h' = t e Th hence by (4.3.3), h' e Th for some / < i, with h' = (st?_ Д
q > 1. Hence, (Л')" = t(_ x > t, = h" by (0.3.4). Hence, (4.1.2) and (4.1.3) hold
(with L in place of H).
Conversely, suppose that (4.1.2) and (4.1.3) hold. We may find a finite
closed subset £ of М(Л) containing h = (h', h"). Taking the notation of (0.3.3)
and (0.3.4), we have to show that he L. We have h', h" e L with h' < h".
Hence, h' = t,, h" = tj with n > i > j.
Suppose that h' e A. Then h' e To, and t0 > • • • > tj > tt = h', hence,
h' e T0\{t0,..., tj}. By (4.3.2), we deduce h’eTj+l. This implies h =
(h', tj) e Tj+ p by definition of this set. Thus h e L by (4.3.5).
Suppose now that h’ ф A. Since h' = t{eTi by (0.3.3), we have by (4.3.3),
h' = (ttg-^e Tk for some к < i, p > 1, t e Tk_ i\tk_ P By assumption, we have
tk_! = (h')" > h" = tj, hence, к — 1 < j. Thus h' e Tk with к < j 4- 1. If
к = j 4- 1, then h' e Tj+1. Otherwise, к < j, hence h' = t,- is not in
4.4 Appendix	101
{tk, tk+1,..., tf, hence, by (4.3.2), h' e Tj+ j also. Thus h = (h', tj) e Tj+l, by
definition of this set, and we conclude by (4.3.5) that he L.	□
4.4	APPENDIX
4.4.1 Optimality of the Hall bases
The construction of the bases of the free Lie algebra presented in this chapter
is optimal, in some sense. Indeed, the following result, due to Viennot (1978,
Theorem 1.1.2), holds: let H be a totally ordered subset of M(A), containing
A, and such that (4.1.2) and (4.1.3) hold; define the Lie polynomials Ph (he H)
by: Pa = a if aeA and Ph = [Ph., Ph..J if h = (h’,h”). Suppose that these
polynomials form a basis of the free Lie algebra. Then H satisfies (4.1.1).
We sketch the proof, which is divided into three parts.
(i)	Suppose that w in A* has a factorization w = /(/ij ... f(hn), ht e H,
hY > • • • > hn. Then for each word u, uw has a factorization uw = /(tj...
f(tp) with |tp| > |/i„| and tr > • • • > tp, ц e H. Indeed, it is enough to show
this when и = a e A. Then, either a > /i15 and f(a)f(hi)... f(hn) is the
desired factorization, or a < hi which implies by (4.1.2) and (4.1.3) that
ki = (a, hj e H. Then either kr > k2, and the desired factorization is
f(ki)f(h2) • • • f(h„), or ki < h2, and since ht > h2, (4.1.2) and (4.1.3) imply
that k2 = (k^ h2) e H. Continuing in this way gives the result.
(ii)	Each word in A* has a unique factorization as in (i). Indeed, (i) shows
that each word has the desired factorization (take w = 1). To prove
uniqueness, we may assume that A is finite, |Л| = q. Then the numbers of
trees of degree n in H is, by assumption, equal to the number in Corollary
4.14. The proof of this result then shows that the number of tuples
(hv ..., hp), h( e H, hi >•••> hp, of total degree n, is equal to qn. As this is
the cardinality of An, the result follows.
(iii)	Suppose, by contradiction, that for some tree h in H, one has h > h".
Denote by r(t) the immediate right subtree of t. Then, for some p > 1, we
have rp(h) = a e A. If a < h, then by (4.1.2) and (4.1.3), (a, h) e H. If on the
contrary a > h, then for some m in {l,...,p}, we have rm(h) > h and
rm~1(h) < h (because rp(h) > h and h” = r(h) < h); then, by (4.1.2) and (4.1.3),
we have (rml(/i). h)e H (because rm~1(h)" = rm(h) > h). In both cases, we
find a nontrivial right factor v of f(h) such that vf(h) e f(H). Hence, by (i),
the word f(h)f(h) has a factorization /(/ij... f(hn), hteH, hx>    >hn,
with \h„\ > |p| + \h\ > |h|. This contradicts the uniqueness in (ii).
4.4.2 Elimination in free partially commutative Lie algebras
Let в be a subset of A x A, not intersecting the diagonal. The free partially
commutative monoid M(A, в) is defined as the quotient of A* by the
102
4 Hall bases
congruence = generated by the relations ab = ba for (a, b) e 0. Similarly, the
free partially commutative Lie algebra L£(A, 0) is the quotient of &(A) by
the Lie ideal 1 generated by the Lie polynomials [a, b], for (a, b) e 0. The
monoid M(A, 0) and the Lie algebra ^(A, 0) are clearly free, each in the
appropriate category. The following result, due to Duchamp and Krob
(1992a), extends the Lazard elimination method (Theorem 0.6).
For m in M(A, 0), denote by TA(m) the set of letters b in A such that
me M(A, 0)b. Note that if (a, b) e 0, then we have for any Lie polynomial:
ad(a) о ad(b)(P) = ad(b) > ad(a)(P) mod I. Indeed, by Jacobi’s identity, ad is
a Lie homomorphism, so that ad(a) ° ad(b) — ad(b) о ad(a) = ad([a, b]). From
this we deduce that и = v implies ad(u)(P) = ad(v)(P) mod 1. Thus ad(x) is
a well-defined endomorphism of £L(A. 0), for any x in M(A, 0). With these
notations, the elimination result is the following. Let C £ A such that
0 n В x В = 0, where В = 4\C. Then the К-module £f\A, 0) is the direct
sum of 0 n C x C) and of the Lie ideal J generated by B. Moreover,
J is freely generated, as a Lie algebra, by the elements ad(m)(b), where m is
any element of the submonoid of M(A, 0) generated by C, and b in В is
such that {b} = TA(mb). For this and related results, see Schmidt (1990),
Duchamp and Krob (1992b, c, 1991a,b, 1992J), and Duchamp (1989).
4.4.3 Another rewriting rule
There is another rewriting rule for sequence of Hall words, which plays the
same role as the relation -> of Section 4.1. It is due to Schiitzenberger (1958,
1986), works only for Hall sets H with the further property
h < h',
(4.4.1)
for any h = (h’, h") in H, but has the advantage of being local. Instead of
standard sequences, one considers sequences (4.1.4) with the property that
for i = 1,..., n — 1, b, < bi+1 implies (h,, bi+1) e H. Instead of legal rises,
one considers rises (b,, hi+1) such that either i + 1 = n, or hi+ j > hi + 2. Then
Theorem 4.3 holds for this new rewriting rule. Note that property (4.4.1)
also appears in Theorem 5.16(vi).
4.4.4 Free Lie superalgebra
Let A be an alphabet with a mapping у: A x A K, where К is a field of
characteristic 0 such that /(a, b)/(b, a) = 1 for a, b in A. For finely homo-
geneous polynomials P, Q, one defines
X(P,Q) = П x(a,b)d^Pde^Q.
a. be A
4.5 Notes
103
Then one defines a bilinear operation on K<A> by
IP,QJx = pq-AP,<2)QP.
Denote by ^Z(A) the smallest subspace of containing A and which
is closed under this operation. The previous construction is due to Ree (1960),
who gives a generalization of the Friedrichs, Dynkin-Specht-Wever and
Poincare BirkhofT Witt theorems and the Witt formula. The construction
of Hall bases extends to УХ(А) as follows (Melancon 1991). Let H be a Hall
set in M(A). Let x(t15t2) = /(Ж),/(t2)X for any tree t15 t2 in M(A). Let
H_ = {he H | h) = — 1}, and define
Hx = H<j {(h, h) | heH_}.
Extend the order of H to Hx by h > (h, h) > к if h e H. к e H and h> k.
Then each word has a unique factorization
/(^.../(hj, hieHx,h1>-->hn,
where each h in H_ appears at most once. Defining Ph, for h in Hx, as in
Section 4.2, the set of polynomials Phl... Phn, with the same condition as
above, forms a basis of К<Л>. Moreover, the Ph, heHx, form a basis of
£fx(A). See also Mikkalev (1986).
4.5	NOTES
The story of Hall bases of the free Lie algebra is not a simple one. They
appear in a paper of M. Hall (1950), with condition (4.1.1) replaced by the
stronger condition that the order be compatible with the degree. However,
similar constructions of ‘basic commutators’ in the free group had already
been done by P. Hall (1933) and Magnus (1937). M. Hall’s construction was
generalized by Meier-Wunderli (1951), Witt (1956), and Schiitzenberger
(1958), by weakening his degree condition. The present condition (4.1.1) was
shown to be sufficient to give bases of the free Lie algebra by Viennot (1978);
this condition was also known to Shirshov (1962) and Michel (1974).
Actually, Viennot shows that this condition is in some sense optimal (see
Section 4.4.1). Condition (4.1.1) is so general that it includes the Lyndon
basis (which is a basis of the free Lie algebra constructed by Viennot (1978)
and Lothaire (1983) by following the lines of the commutator calculus of
Chen et al. (1958)) and the Shirshov basis (1958); this was not the case with
the original bases of M. Hall.
We warn the reader that there are four symmetries which may change the
presentation of Hall bases: one can reverse the words and also reverse the
order. For instance, our presentation is compatible with that of Viennot
104	4 Hall bases
(1978) after these two reversals. We have chosen this presentation to make
it compatible with the Lyndon basis of Lothaire (1983).
Theorem 4.3 is due to Melancon (1992), who extended a method of
Schiitzenberger (1958), itself related to the collecting process of P. Hall
(1933); see also M. Hall (1959). It allows one to quickly prove Corollaries
4.4, 4.5, and 4.7, which were known to all the previous authors. The proof
of Theorem 4.9(ii) follows Melancon and Reutenauer (1989), and that of
part (i) of this theorem, together with the underlying algorithm, follows
Schiitzenberger (1986). The assertion on the Lie polynomials in Corollary
4.14 is due to Witt (1937) and Corollary 4.15 is due to Schiitzenberger (1958).
The fact that the bases obtained by Lazard are the same as the generalized
Hall bases (Theorem 4.18) is due to Viennot (1978).
Other bases of the free Lie algebra were constructed by Kukin (1978) and
Blessenohl and Laue (1990b), the latter by purely group theoretic methods.
See also Corollary 8.20. A formula similar to that of Corollary 4.14 appears
in Witt (1956) in the case of the free Lie p-algebra.
5
Applications of Hall sets
We begin by introducing Lyndon words and the Lyndon basis of the free
Lie algebra; this is a particular Hall set, with special properties, such as
triangularity of the change of basis. The dual basis of the Poincare-Birkhoff-
Witt basis associated to a Hall basis may be computed recursively, by using
the shuffle product; this is done in Section 5.2. In the next section a special
Hall basis is constructed, which is compatible with the derived series of the
free Lie algebra. The special properties of Lyndon words, with respect to the
alphabetical order, are also true for Hall words once the appropriate order
on words is defined: to obtain it, one factorizes each word, and then orders
sequences of Hall words alphabetically (Section 5.4). In the final section, we
show how Hall sets lead to the construction of variable-length codes, with
various synchronization properties.
5.1 LYNDON WORDS AND BASIS
Let A be a totally ordered alphabet. We order the free monoid A* with
alphabetical order, that is, и < v if and only if either v = их for some
nonempty word x, or и = xau', v = xbv for some words x, u', v' and some
letters a, b with a < b. Observe that this order on words is simply the order
in which they would appear in some dictionary.
A Lyndon word on A* is a nonempty word which is smaller than all its
nontrivial proper right factors; in other words, w is a Lyndon word if vv / 1
and if for each factorization w = uv with u, v Ф 1, one has w <v.
Theorem 5.1 The set of Lyndon words, ordered alphabetically, is a Hall set.
The corresponding Hall basis has the following triangularity property: for each
word w = f ... ln written as a decreasing product of Lyndon words, the
polynomial Pw = Ph ... Pln is equal to w plus a Z-linear combination of greater
words.
We need some properties of the alphabetical order, gathered in the next
lemma.
106	5 Applications of Hall sets
Lemma 5.2 (i) If и < v and и is not a prefix of v, then их < vy for all words
x, y.
(ii) If и < v < uw, then v = uv' for some word v' such that v' < w.
Proof (i) This is an immediate consequence of the definition of the order.
For (ii), imagine the words u, v, uw written in a dictionary, and the assertion
immediately follows.	□
Proof of Theorem 5.1 (a) Define the standard factorization of each word
w of length >2 to be the factorization w = uv, where v is the smallest
nontrivial proper right factor of w for the alphabetical order. Then define
recursively for each nonempty word w a tree t(w) in M(A) by t(d) = a if a
is a letter, and t(w) = (t(u), t(v)) if w = uv is its standard factorization.
Evidently, the foliage of r(w) is w, so w i—> t(w) is injective and the set
{t(w)|we4*} inherits the alphabetical order. We show that the set
{t(w) | w Lyndon} is a Hall set.
In view of eqns (4.1.9)-(4.1.11), it is enough to prove the following two
assertions.
If w is a Lyndon word with standard factorization uv,
then u, v are Lyndon words with и < v, w < v, and either и is a letter, (5.1.1)
or the standard factorization of и is xy with у > v.
If u, v are Lyndon words with и < v, then uv is a Lyndon word. (5.1.2)
Let us prove (5.1.1): we have и < uv = w < v, the first inequality by
definition of the order, and the second because w is Lyndon. So и < v, w < v.
Moreover, v is smaller than all its nontrivial proper right factors, because
these are nontrivial proper right factors of w and v is by definition the smallest
among them. Hence г is a Lyndon word.
Let у be any nontrivial proper right factor of u. Then yv > v. because v is
the smallest nontrivial proper right factor of w. If we had у < v, then Lemma
5.2(ii) and the inequalities у < v < yv would imply v = yv' for some word v'
with v' < v, a contradiction (because v' would be a proper nontrivial right
factor of w = uv = uyv'). Hence, у > v.
This shows at the same time that и is a Lyndon word (because и < v < y),
and that if и = xy is its standard factorization, then у > v.
Hence, (5.1.1) is proved. In order to prove (5.1.2), let s be a proper
nontrivial right factor of uv. Then we have three cases.
(i)	5 is longer than v, hence 5 = u'v for some nontrivial proper right factor
u' of u. Since и is Lyndon, we have и < и', and since и is not a prefix of u',
we have by Lemma 5.2(i), uv < u'v = 5.
(ii)	s = v: if и is a prefix of v, then v = uv'; since v is Lyndon, we have
5.1 Lyndon words and basis	107
v < v', hence uv < uv' = v; on the other hand, if и is not a prefix of v, then
from и < v and Lemma 5.2(i), we deduce uv < v.
(iii)	5 is shorter than v: then 5 is a nontrivial proper right factor of v, hence
v < s (because v is Lyndon); since by (ii) uv < v, we deduce uv < s.
This proves that uv < s, hence also (5.1.2), and we conclude that the set
of Lyndon words is a Hall set.
(b) Note that if / is a Lyndon word written as a nontrivial product / = uv,
then by definition / < v; since v < vu, we have / < vu.
We show first that for each Lyndon word /, the corresponding Lie
polynomial Pt is equal to / plus a Z-linear combination of greater words of
the same length as /. This is clear for I = a e A, because Pa = a. If |/| > 2, let
I = uv be its standard factorization; then u, v are Lyndon words and by
induction
Pu = w + E ax^ pv = v + E РУу-
x>u	у > V
Then, by definition of P(, we have
P^lPu^l
= PUPV - PVPu
= uv + e РуиУ + E axxy + E ахРухУ -vu - E a*vx
y> V	x>u	x>u	x>u
y>v
- E РуУи - E ахРУух.
y> V	x> и
У> V
Then, the assertion follows from the inequalities I = uv < vu and
U < X, V < у => uv < uy, UV < XV < xy,
and, similarly,
и < x, v < у => vu < vx, vu < yu < yx.
In these implications, we have used the following trivial fact: r < s, r’ < s',
|r| = |r'|, |s| = |s'| => rs < r's', with strict inequality if one of the first two
inequalities is strict.
The latter fact (extended to arbitrarily long products) will also imply the
theorem; indeed, take w as in the statement of the theorem. Then
P(l = /.+ E *x<-
x,>l,
|X,| = |I.|
where * denotes coefficients in Z, whose exact values are not important. Thus
Pw = fl Р,. = ^---1п + ^У1-- Уп
: — 1
108
5 Applications of Hall sets
where the summation runs over words y15 ..., yn with y, >	|y,-| = |/,-| and
у j > Ij for at least one j; in this case, yx ... yn> f ... ln = w. This proves the
theorem.	□
5.2 THE DUAL BASIS
We now return to arbitrary Hall sets. By Theorem 4.9(ii) the decreasing
products of Hall polynomials Phi ... Phn (h1>    > hn) form a Z-basis of
Z</1>. By Corollary 4.7, each word w in A* has a unique decreasing
factorization into Hall words w = h{ ... hn (hr >    >hn}, thus, we define
Pw = Phl-.Phn
and we obtain a well-defined basis (Pw)weA* of the Z-module Z</1>, indexed
by words.
In this section, we shall consider the dual basis. Recall that by Section 1.1,
the dual space of is canonically isomorphic to	Thus the dual
basis of the basis (Pw)weA* is the family (Sw)weA* of formal series such that
for any word и
U= E (S„,u)P„.	(5.2.1)
we A*
Note that, since Pw is a finely homogeneous polynomial of same partial
degrees as w, so is Sw. Moreover, by Lemma 4.11, the following result holds:
let и = аг ... ap (a{ e A) and w = hr ... hn (ht e H, hr >    > hn); then
(Sw, u) is equal to the number of leaves labelled (hl,..., hn)
in a derivation tree of(a15..., ap).
See Examples 4.10 and 4.12.
We shall give formulas for the series Sw. It will be more convenient to
work in	although the series have coefficients in Z; this could be
avoided by defining a structure of algebra with divided powers in the shuffle
algebra Z<</1>>.
Recall that the shuffle product ш has been defined in Section 1.4.
(5.2.2)
Theorem 5.3 (i)	= 1.
(ii) For any Hall word h = av (a e A), one has Sh = aSv.
(iii) For any word w = h\'... hk written as a decreasing product of Hall
words (/q > • • • > hk, f,..., ik e N), one has

5.2 The dual basis
109
We need a lemma. Denote by => the relation on standard sequences, which
is defined by 5 => t if either t = 2f(s) or t = pt(s) for some i; see eqns (4.2.3)
and (4.2.4). Denote by => the transitive closure of =>. Derivation trees are
defined in Section 4.2.
Lemma 5.4 (i) Let s be the standard sequence (7i15..., hn) with n>2 and
suppose that hr > h( for i = 2,..., n. Then for any standard sequence t with
s^> t, t is of length at least 2.
(ii) Let s be the standard sequence (h^...,^) with h2>   - >hn. If
ht... hn is a Hall word, then there is exactly one chain s =>•••=> (ht ... hn).
If s^> t and t (7ц ... hn), then t is of length at least 2.
Proof (i) By assumption, for any rise h{ < hi+1 in 5, one has i > 2. This
implies that Af(s) and p,(s) are of length >2. Moreover, they satisfy the same
condition as s, that is, their first element is greater than or equal to the
others. This is clear for p,-(s), and for z/s), note that by eqns (4.1.7) and
(4.1.10), hi+l > hihi+l. Hence, (i) follows by induction on the length of the
chain s=> t.
(ii) If и = 1, there is nothing to prove. Suppose n > 2. If hx > h2, then 5
is decreasing and so there is no nontrivial chain starting from s. Moreover,
7ц ... hn is not a Hall word, by Corollary 4.7.
So we may assume that 7ц < h2. This is surely the only rise, so that
2,(s) = (h2, hi, h3,..., h„) and p,(s) = (h^, h3,..., 7i„). Note that h2 is the
greatest element of x;(s), so by (i), s => t implies that t is of length at least 2.
Moreover, pfs) is shorter than s and satisfies the same hypothesis. So we
may conclude by induction.	□
Proof of Theorem 5.3 (i) This is clear because = 1 and the other Pw are
homogeneous of degree > 1.
(ii) Let h = av be a Hall word with first letter a. Since Sh has no constant
term, the equality Sh = aSv is equivalent to saying that for any word и and
any letter b, (Sh, bu) = batb(Sv, u). We have by (5.2.1)
w = E <s*- u)p„-
w
Hence,
bu = £(sw,u)bpw.	<5-2-^
w
Choose a word w and write it as a decreasing product of Hall words,
w = hi ... hn. Then the sequence s = (b,ht,... ,hn) is standard, and by
Lemma 4.11
bPw = bPln...Ph, = PW = Y.«,pW-
НО	5 Applications of Hall sets
where the summation is over all decreasing sequences t of Hall words and
where a( is the number of chains s t in a fixed derivation tree of s. By
Lemma 5.4(h), each t is of length >2, except when bw = bhr ... hn is a
Hall word, in which case there is exactly one chain from s to (bw).
When we put this into (5.2.3), we obtain
bu = Y, (Sw, u)Pbw + sum °f decreasing products of length >2
bwHaiiword	of Hall polynomials.
Hence, the coefficient of Ph = Pav in this sum is equal to 0 if a b, and to
(Sv, u) if a = b. In other words, (Sav, bu) = 3a b(Sv, u), as required.
(iii) Note that, by definition of the dual basis, we have (Sw, Pu) = dw u. In
particular, if w is a Hall word and и is not, then (Sw, Pu) = 0.
Write w = Wj ... w{ as a decreasing product of Hall words, hence
i = ц 4- • • • + ik. We evaluate (SW1 lu- • -ш SWi, Pu), which is equal to
(SWI ® • • • ® SWi, bi(Pu)) by Proposition 1.8. By (1.5.6)
3i(P) = P®1®---®1 + 1®P®---®1+--+1®1®---®P,
(5.2.4)
for each Lie polynomial P.
Write и = Uj ... un as a decreasing product of Hall words. Then
Pu = PUl... PUn. Now, 3; is an homomorphism and each PUj a Lie polynomial.
Hence,
Wu) =	(5-2.5)
By inspection of (5.2.4) and (5.2.5), we find that f(Pu) is a sum of terms of
the form Qr ®  • • ® Qt, hence that (SWI ш •   ш SWi, 3((Ри)) is a sum of terms
of the form (SWI, Qt)... (SWi, Q,): if i > n, then in each term, at least one Qj
is equal to 1, hence, since (SWi, 1) = 0, we have (SWi ш- • -ш SWi, Pu) = 0; if
on the contrary, i < n, then in each term, at least one Qj is a decreasing
product Pu = P„ ... PUir with r > 2, hence, since (SWt, Pu.) = 0, we also have
(SW1 ш • • • ш Sw , Pu) = 0. In the remaining case i = n, we obtain, because
(SWi, 1) = 0,
(SWI ш • •  ш SWn, Pu) = £ (SWI, PUaJ ... (SWn, PUaJ
aeS„
W1 , Ufr( 1 j ...	, Urrfn } *
<reS„
If w / u, then (w15..., vv„) (ub..., un), and since both sequences are
decreasing, the right-hand side vanishes. If w = u, then by Corollary 4.7
(w15..., w„) = (iq,..., u„); since (wp ..., w„) = (7q, ..., 7ц,..., hk,..., hk),
each hj repeated ц times, the right-hand side is equal to the number of
5.2 The dual basis
111
permutations fixing the previous sequence, that is, ij ... ik!. Finally,
— ,№7
. ,ik!
ш---ш^,Ри) = ^,
which proves (iii), by definition of the dual basis.
□
Corollary 5.5 The shuffle algebra is a free commutative algebra over
the set Sh (h Hall word).
Proof The families (Sw) and (Pw) are dual bases. Hence, besides eqn (5.2.1),
we also have the dual relation, for any polynomial Q
c= Z (p»,e)s„.
we A*
This shows that the polynomials Sw form a basis of the space <0><Л>. Because
of Theorem 5.3(iii), this implies that the polynomials Sh (h Hall word) form
a free generating set of the shuffle algebra 0»<Л>.	□
In the next result, we again use the algebra .2/ introduced in Section 1.5.
Corollary 5.6 The following identity holds in the algebra sd
E W® w = П exp(Sh® Ph),
weA*	heH
where the product has to be taken in decreasing order.
This result could also be stated in the algebra End(Q<4>) with the
convolution product * (see Section 1.5): it gives a formula for the identity
as a product of exponentials of elementary endomorphisms.
Proof The right-hand side is
h \i>oi!	/ hl >    > hk ll-  • • lk-
ij..ik > 1
Because of the definition of PH, and Theorem 5.3(iii) this is
E sw®p...
we A*
5 Applications of Hall sets
112
which is equal to
Eu ® E u)pw I = e u ®u
by (5.2.1).
□
5.3 THE DERIVED SERIES
Define a sequence of subspaces of £f(A), called the derived series, in the
following way:
= ^(Л), £Fn +1 =	<£"].
The latter means that ^fn+1 is the subspace of £\A) generated by the
polynomials [P, Q] for P, Q in <£n. Each subspace P£n is a Lie ideal of ^(Л),
and in particular ^fn+1 c P£n-, this is an easy consequence of the Jacobi
identity, left as an exercise.
We define a particular Hall set H which will be shown to be compatible
with the derived series. Define Ho = A and order it totally; now define
recursively Hn+1 as the set of trees of the form
h = (.. .((hr, h2), h3),..., hk),	(5.3.1)
where к > 2 and where ht,..., hke Hn with
/ii < h2 > h3 > • • • > hk.	(5.3.2)
Now order Hn+l totally.
Finally, let H = (Jn>0Hn and extend the order of the Hn to H by the
condition
h e Hn, к e Hp, n < p => h > k.
Symbolically, this may be written as H0>H1>
Equation (5.3.1) is illustrated in Fig. 5.1.
(5.3.3)
>Hn>Hn+1>
Theorem 5.7 The set H is a Hall set. For each n > 0, the set of Hall
polynomials {Pw | w e ljp>„ Hp} is a basis of У?п.
Proof (a) In order to prove the first assertion, we have only to verify
conditions (4.1.1)—(4.1.3).
Suppose h e H is of the form fh’, h"). Then h is in Hn+l for some n > 0,
and is of the form (5.3.1). Then h" = hk is in Hn, so h < h" by (5.3.3), and
(4.1.1) holds.
5.3 The desired series
113
If к > 3, then h' = (.. .(hv h2),... ,hk_l) is in Hn+l. Hence, h' < h" by
(5.3.3). Moreover, h' = (x, y) with у = hk_r, and by (5.3.2), у > h". So (4.1.2)
and (4.1.3) hold in this case.
On the other hand, if к = 2, then h' g Hn and h' = hx < h2 = h" by (5.3.2),
and (4.1.2) holds. Moreover, if h’ is not a letter, then n > 1 and h' = (x, y)
must also be of the form (5.3.1), so that у is in Hence, у > h" by (5.3.3),
and (4.1.3) holds.
Suppose now that (4.1.2) and (4.1.3) hold for h = (h', h"). We have to show
that h is of the form (5.3.1) and that (5.3.2) holds. If h', h" are in the same
Hn, then h is of the form (5.3.1) and (5.3.2) holds with к = 2, because h' < h"
by (4.1.2). Otherwise, because of (4.1.2) and (5.3.3), we have h' g Hn+l, h" g Hp
with n + 1 > p. Then h' is of the form (5.3.1) and condition (5.3.2) holds.
Moreover, writing h' = (x, y), we have у > h" by (4.1.3), and у = hk by (5.3.1).
Since hk is in H„ and h" in Hp, we deduce from (5.3.3) that n < p. This,
together with p < n + 1, shows that p = n. Then we set h" = hk + l, so h is of
the form (5.3.1), and condition (5.3.2) holds (with к + 1 instead of k).
(b) Define the level l(t) of a tree t inductively by /(t) = 0 if t g A and,
otherwise, t = (t', t"), and let
fl + /(t') if I(t') = /(*");
tsup(/(t'), I(t")) if/(f) * l(t").
114	5 Applications of Hall sets
The following facts are easy to verify:
if s is a subtree of t, then l(s) < l(t);	(5.3.4)
if ti is obtained from t by replacing its subtree s by Sj and if)
/(5) < /(sj, then l(t) < 1(h).	J
We claim that Hn is the set of trees in H which are of level n. Indeed, note
first that, by definition of the level, a tree which is not reduced to a single
letter is of level > 1. Hence, Ho is the set of Hall trees of level 0. Arguing by
induction, let h be in Hn+ p hence of the form (5.3.1); then h b..., hk are in
Hn, hence of level n by induction; an immediate induction on к (starting from
к = 2) and the definition of the level shows that h is of level n + 1. This
proves the claim.
Note that the second assertion of the theorem is true for n = 0, by Theorem
4.9(i). The general case will follow by induction if we prove the following.
Let h, к be Hall trees of level >n; then [Ph, Pk] is a linear combination of
Hall polynomials Pt, with t of level >n + 1.
Note that under the previous hypothesis the tree (h, k) is of level > n + 1.
Thus, in view of the algorithm stated after the proof of Theorem 4.9 it is
enough to show that each step of this algorithm, when applied to a tree t,
produces only trees of level > l(t). This is clear for step (4.2.8) and (4.2.9).
For step (4.2.10), we have s = (s', s"), where s' = (x, y) and s" are Hall trees
and x < у < s", s' < s'. By the claim and (5.3.3), we have /(x) > l(y) > l(s")
and /(s') > /(s"). If we had /(s') = /($"), then we would deduce /(s') = l(s") <
l(y) < /(x) < /(s') (the latter inequality by (5.3.4)), hence, equality everywhere;
since s' = (x, y), this would imply, by the definition of the level, /(s') =
/(x) + 1 > /(x), a contradiction. Thus we have /(s') > l(s") which implies
l(s) = /(s'), by definition of the level. Now, by (5.3.4) and (5.3.5), /(((x, s"), y)) >
/((x, y)) = /(s') = /(s), and /((x, (y, s"))) > /((x, y)) = /(s') = /(s). Hence, the
trees tj and t2 of step (4.2.10), obtained by replacing the subtree s of t by
((x, s"), y) and (x, (y, s'')), respectively, are by (5.3.5) of level >/(t). This
proves the theorem.	□
5.4 ORDER PROPERTIES OF HALL SETS
Let H be a Hall set in A*. According to Corollary 4.7, each word w in A*
has a unique factorization w = hr .. .hn, hte H, hr >• - >hn. We use this
bijection between A* and the set of decreasing sequence of Hall words to
define a total order on A*, which extends the order on Hall words, and which
we therefore still denote by <. This order is obtained by carrying to A* the
alphabetical order on the sequences of Hall words. This means that if z is
another word in A*, factorized as z = kr ... km, kte H, kx >• - >km, then
5.4 Order properties of Hall sets
115
(5.4.1)
w < z if and only if
either n < m and = fc,- for i = 1,..., n,
or for some j, hx = ki,..., hi_l = k^ t and < kt.
In the next result, H is a fixed Hall set, with the corresponding order < on
Л*, defined in (5.4.1).
Theorem 5.8 Let h be a word in A*. Then the following conditions are
equivalent:
(i)	h is a Hall word;
(ii)	for each nontrivial factorization h = uv, one has h < v;
(iii)	for each nontrivial factorization h = uv, one has h < vu.
We need several lemmas. The first one has a geometrical interpretation
(see Fig. 5.2).
116	5 Applications of Hall sets
Lemma 5.9 Let h be a Hall word and h = uv a nontrivial factorization. Then
there exist Hall words kr,..., km, ht,... ,h„ such that
u = kx...km, v = hl...hn,
and kt,..., km < ht, ht > •  • > hn> h". Furthermore:
(i) if v is longer than h", then hr > (h')" and n > 2;
(ii) if v is shorter than h", then hn > (h")".
Proof Let h = uv, as in the lemma. Then h = f(t), for some Hall tree t. We
prove by induction on \h\ that
u =f(si). ..f(sm), v = f(tj ...f(tn)
for some Hall trees s19..., sm, tl,...,tn with s1?..., sm < tx, >•••>!„> t".
We have t = (f, t") where t', t" are Hall trees by (4.1.2), hence we have
ЛОЖ) = uv.	(4.2)
It may happen that и = f(tf and v = f(f), in which case we take m = n = 1
and Sj = t', tj = t". Then (4.1.2) gives us and we are done. Otherwise,
we have two cases, according to the relative lengths of v and h" = f(t").
In case 1, v is longer than h". Then by (5.4.2), v = xf(t"), f(t') = их, for
some nonempty word x. By induction, applied to the Hall word t', we obtain
u = /(sj ... f(sm), x = f(ti)...f(tn-i),
for some Hall trees sl,...,sm, tl,...,tn_l such that s19..., sm < tx,
G > • •  > t„_ i > (t'K and n > 2. We have (t')" > t" by (5.1.3), hence
tn-i > t", and we may take tn = t". Moreover, (i) holds.
In case 2, v is shorter than h". Then by (5.4.2) for some nonempty word
x, f(t") = xv, и = f(t')x. Then, by induction applied to the Hall word t", we
obtain
x = f(s2)... f(sm), v = f(tf)... f(t„),
for some Hall trees s2,..., sm, t1,...,tn with s2,..., sm < t19 tx > • • • > tn >
(t")". By (4.1.1), we have (t")" > t", and by (4.1.2), t" > t'. Hence, t' < tt, and
we may take = t'. Moreover, (ii) holds.	□
Recall that standard sequences of Hall words (or trees—we identify once
more Hall words and trees—see Corollary 4.5) were defined in Section 4.1,
where the binary relations -> and its transitive closure are also defined.
We denote by w(s) the word w(s) = /ц ... hn, for a sequence of words
s = (hi,..., hn), and by max(s) the greatest word in the sequence.
Lemma 5.10 Let s = (hi,..., h„) be a sequence of Hall words. Then there
5.4 Order properties of Hall sets	117
exists a standard sequence t = (kt,... ,km) of Hall words such that w(s) = w(f)
and max(s) = max(t). Moreover, max(s) = hY if and only if max(t) = kr.
Proof (induction on <5(s) = j |/i{| — n). If s is standard (in particular if
<5(s) = 0), there is nothing to prove. Otherwise, there are some i, j such that
h" < hj. Replace in s the word by the sequence (Jih h"), and call the new
sequence s'. Evidently, w(s') = w(s). Note that /i; < h" by (4.1.1), hence hi < hj
which shows that /i; max(s); since by (4.2), /1- < h", we deduce that
max(s) = max(s'). Moreover, if max(s) = ht, then i 1, hence is the first
element of s'; conversely, if max(s) hu then either i ± 1, and hr is the first
element of s', and not its maximum because max(s') = max(s); or i = 1, and
h\ is the first element of s' with h\ < h’[ (by (4.1.2)) <hj < max(s) = max(s'),
and h\ is not the maximum of s'.
Now, <5(s') = <5(s) — 1, which allows us to conclude by induction. □
Lemma 5.11 Let s = (ht,..., hn), t = (kv ..., km) be standard sequences
such that s —+ t. Then
(i)	max(s) > max(t);
(ii)	if max(s) = hr, then hr = кг = max(t);
(iii)	if max(s) Ф hY and t is decreasing, then max(t) < max(s).
Proof (i) This is a consequence of (4.1.7) and of the inequality
(/ii,/ii +1) < hi +1, a consequence of (4.1.1).
(ii) In this case, (ht, h2) is not a rise, so that hr still appears as the first
element of t', for any t' with s -> t'. Hence, (ii) follows from (i), by induction.
(iii) Let s = s0 -> st ->----> sn = t. Let h = max(s), and denote by g(sf
the first element of sf. We show by induction on i that g(sf < h. This is
true for i = 0, by hypothesis. Suppose it is true for sf; then either
g(si + J = g(sf < h; or g(si + 1) is by (4.1.7) the concatenation kl of the
two first elements of s{, hence, by (4.1.1), g(si+l) < I < тах(х,) < max(s0)
(by (i)) = h.
The previous argument shows that g(t) < max(s). Since t is decreasing,
max(t) = g(t) and we are done.	□
Lemma 5.12 Let h be a Hall word of length >2. Then h" is its smallest
nontrivial proper right factor.
Proof Let v be a nontrivial proper right factor of h. If v is longer than h",
then by Lemma 5.9 we have v = hx ... hn for some Hall words ht,..., hn
with hr > • • • > hn > h" and n > 2. Hence, we deduce from (5.4.1) that v > h".
If v is shorter than h", then by Lemma 5.9 we have v = h{ ... hn for some
Hall words hr,... ,hn with ht > • • •> hn> (h")". Since hY > (h")" > h" by
(4.1.1), we deduce from (5.4.1) that v > h".	□
118	5 Applications of Hall sets
Proof of Theorem 5.8 (i) => (ii) This is a consequence of (4.1.1) and Lemma
5.12.
(ii) => (i) Suppose w is not a Hall word. Then by Corollary 4.7 we have
w = hi ... hn for some Hall words й19..., hn with hr > • • • > hn and n > 2.
Hence, by (5.4.1), the right factor hn of w is smaller than w, because hn <
(i) => (iii) By Lemma 5.9, we have и = kr ... km, v = hY ... hn for some
Hall words kt, hj which are all <hr and such that hY > h". Hence,
vu = hi ...h„ki ...km.
By Lemma 5.10, there exists a standard sequence of Hall words t = (/p ... ,lp)
such that vu = f ... lp and hY = f = max(/t). Hence, Theorem 4.3(iii) and
Lemma 5.11 (ii) imply that t t', for some decreasing sequence of Hall words,
such that the first element in t' is hp We have hi > h" > h, by (4.1.1), so that
by (5.4.1), vu > h.
(iii) => (i) Let w be a word which is not a Hall word. Then by Corollary
4.7 we have w = ht ... hn for some Hall words ht with hr >  •  > hn, and
n > 2. If hi = hn, then these Hall words are all equal and w = hnhi ... hn_i.
Otherwise, we have hn < hr. Let w' = hnhi ... hn_i. Then, by Lemma
5.10, applied to s = (hn, hr,..., hn_ J, we may find a standard sequence
t = (ki,..., fcOT) such that w' — kr .. . km and hi = max(s) = max(t) > kv By
Theorem 4.3(iii) and Lemma 5.1 l(iii), we find a decreasing sequence t' of
Hall words such that t t' and max(t') < max(t) = hr. This implies by (5.4.1)
that w' < hi < w.
Hence, we have found in all cases a nontrivial factorization w = uv such
that w > uv.	□
The next result shows that the standard factorization of a Hall word, and
the factorization of a word into a decreasing product of Hall words, may be
found with the order < on A* introduced above.
Theorem 5.13 (i) Let h be a Hall word of length at least 2 and h'h" its
standard factorization. Then h" is the longest proper right factor of h which
is a Hall word, and also the smallest nontrivial proper right factor of h.
(ii) Let w be a word and w = hi ... hn be its decreasing factorization into
Hall words. Then h„ is the longest right factor of w which is a Hall word, and
also the smallest nontrivial right factor of w.
Proof (i) Let г be a proper right factor of h, longer than h". Then by Lemma
5.9 the decreasing factorization of v into Hall words is hr ... hn, with n > 2.
Hence, v is not a Hall word.
Now, the remaining assertion follows from Lemma 5.12.
(ii) Let у be a right factor of w, longer than hn. Then for some
i = 1,..., n — 1, у = vhi+i ... hn where г is a nontrivial right factor of h,. If
5.5 Synchronous codes	119
v = hh then у is not a Hall word, and by (5.4.1), у = h{... hn, because й,- > h„.
If v is a proper right factor of hh then by Lemma 5.9, у = k} ... kmhi+1... h„
with kt > > km > h" > h{ > hi+l > • •  > hn, by (4.1.1). Hence, again, у is
not a Hall word and у > hn by (5.4.1).
Now, let у be a nontrivial right factor of w, shorter than hn. Then у is a
nontrivial proper right factor of hn, so that hn < у by Theorem 5.8(ii).	□
Corollary 5.14 Let h = aw be a Hall word, where a is a letter, w is a nonempty
word, and let w = hr ... hn be the decreasing factorization ofw into Hall words.
Then h" = h„.
5.5 SYNCHRONOUS CODES
A submonoid M of A* has a unique minimal generating set X. Such a set
is called a code if M is a free monoid, hence freely generated by X. In other
words, a set X is a code if whenever ..., xn, yp ..., yp are elements of
X with . x„ = yt ... yp, then n = p and xt = yf for i = 1,..., n. When
X is a code, the monoid homomorphism from the free monoid X* into A*,
which extends the identity on X, is an isomorphism from X* onto M: it
‘encodes’ X* into A*. We shall identify both monoids, and denote M also
by X*.
Let X be a subset of A*, distinct from *1}. If X satisfies
v, uv e X => и = 1,	(5.5.1)
for any words u, v in A*, then X is easily seen to be a code: such a code is
called a suffix code; this is because (5.5.1) states that no word in X is a right
factor of another word in X. It is easily verified that a submonoid M of A*
is generated by a suffix code if and only if M is left unitary, i.e.
r, uv e M => we M	(5.5.2)
for any words u, v in A*. A suffix code is called complete if each word w in
A* is the right factor of some word in X*; in other words, we have for some
words и in A* and y1?..., yp in X\ uw = yt ... yp. This implies that we may
write
w = sxj ... xn, s e S, n > 0, v, e X,	(5.5.3)
where S is the set of proper right factors of the words in X. It is easily seen
that the representation (5.5.3) is unique, since no word in X is a right factor
of another word in X. A word tn in A* is called synchronizing for X if
inveX*	(5.5.4)
for any word г in A*. In particular tn is then in X*. If a word w contains
120
5 Applications of Hall sets
as a factor a synchronizing word m, then the computation of the representa-
tion (5.5.3) is divided into two parts; indeed, we have w = umv, hencemv e X*
by (5.5.4), hence the knowledge of the representation (5.5.3) for и and mv
implies that of w.
Example 5.15 Let A = {a, b}, M the submonoid {1} и {aw | w e A*}. Then
M is generated by the suffix code X = {abn | n > 0}. The set S of proper right
factors of X is {bn | n > 0}. Each word in A* has a unique representation
bn(abn')... (abnk). Each nonempty word in M is synchronizing.
We consider now a Hall set H in A*, and the total order < associated with
it, as in Section 5.4. Recall that each word w in A* has a unique representation
as a decreasing product of Hall words.
Theorem 5.16 Let К be an upwards closed subset of H, i.e. \/k e K.,Vhe H,
к < h => he K. Let M = {w e A* | Vk e K, w < k}.
Then the following properties hold:
(i)	M is a submonoid of A*;
(ii)	M is free, generated by the suffix code X = {he H\K | h e A or h" e K};
(iii)	the subset H\K of X* is a Hall set in the free monoid X*;
(iv)	X is complete if and only if the set
S = {kY ... kn | к, e K, n > 0, kr >• >kn}
is the set of proper right factors of the words in X. This is the case in
particular if К is finite and |Л| > 2.
(v)	if К is of cardinality p, then for each nonempty word m in M, mp is
synchronizing;
(vi)	suppose that H satisfies the following property: for any h in H\A, with
standard factorization h'h", one has h' > h. Then for each nonempty word
m in M, there exists p such that mp is synchronizing. If К Ф H, X is
complete.
Note that, in view of (5.4.1), M is by definition the set of words of the form
hr ... hn, n > 0, A, e H\K, /ц > • • • > hn.	(5.5.5)
Assertion (i) of the theorem is that M contains all the words /ц ... hn with
^Н\К.
Note that Lemma 4.19 is a particular case of the theorem.
Proof (i) By definition (5.4.1) of the order < in A*, the empty word is in
M. Let w, z be two elements in M: then, by (5.4.1), w = /ц ... hn, z = kr ... km
with hi,...,hn, k!,...,kmeH, /ц > • • • > hn, ki>--->km; moreover
kp kj < к for any к in K, by definition of M.
Then wz = hi... h„k! ... km, and by Lemma 5.10 we have wz = f ... lp,
5.5 Synchronous codes
121
for some standard sequence (/19... ,lp) with max(/t) < к for any к in K.
Hence, by Theorem 4.3(iii) and Lemma 5.1 l(i), we have wz =	. mq with
g H, > •  > mq and < к for any к in K. Thus by (5.4.1), we deduce
wz < к for any к in K, i.e. wz e M.
(ii)	By Lemma 5.12, for each element h in X and each nontrivial proper
right factor v of h, one has v > h"; since h" e К and since К is by hypothesis
upwards closed in H, we cannot have vgH\K, and a fortiori not v e X.
Hence, X is a suffix code. It generates M: indeed, M is generated by H\K,
and for h in H\K, either h g X, or h" g H\K, henceh'eH\K (because h' < h"
by (4.1.9) and К is upwards closed) and by induction h', h" e X*, implying
h = h'h"eX*.
(iii)	Let h g H\(K и X). Then h" ф К, which by (4.1.9) implies that h' ф К,
because К is upwards closed. So h', h" g H\K and h' < h". If h' ф X, then
Кф A and (h')" > ft" by (4.1.11).
Conversely, let h,kG H\K with h < к and either h g X or h" > k. We show
that hkGH\K. In the first case, either heA and then hkGH because of
(4.1.11), or h g X\A, hence h" g K, implying h" > к (because к ф К, and К is
upwards closed), thus hkGH by (4.1.11). In the second case, (4.1.11) gives
directly hkGH. Moreover, hk < к by (4.1.10), so that ЬкфК, and finally
hkGH\K.
To conclude that H\K is a Hall set in X*. note that X is contained in
H\K, and H\K is contained in M = X*, by definition of M.
(iv)	Let S be as in the theorem. Then each word w in A* has a unique
decomposition w = sxt ... xn, s g S, n > 0, xtG X. Indeed, this is a con-
sequence of Corollary 4.7, of the definition (5.5.5) of M, and of the fact,
proved above, that M is freely generated by X.
If X is complete, then by uniqueness of (5.5.3), we must have necessarily
that S is the set of proper right factors of words in X.
Conversely, if each word in S is the right factor of some word in X, then
each word in A* is the right factor of some word in X*, because of the
representation sxj ... xn above. Hence, X is a complete suffix code.
If К is finite, then X is complete because of (v) and the fact that M / {1}:
indeed |Л| > 2, hence H is infinite and 0 X H \K M.
(v)	Let К = {kp > • • • > ki}. We show by induction on i that ... /с/1
(Js > 1) is in M. This is enough in view of (5.5.5) and Corollary 4.7. We may
write m = hi ... hn with hr >   • > hn, ht g H\K, n > 1. Then, in view of
Lemma 5.10, there exists a standard sequence of Hall words (^,...,10
such that w = mk{'.. .k{' = .. .lp with kt = max(/7) f. Hence, Theorem
4.3(iii) and Lemma 5.11 (iii) imply that w = pY ... pq for Hall words p19..., pq
with Pi >• - >pq and <kt. Thus, we conclude by induction on i that
ml~1w is in M, as desired.
(vi)	If H has the stated property, then so has the Hall set described in
(iii). Let m in M, m * 1. We show that for p = |m|, mp is synchronizing, by
induction on |m|.
122	5 Applications of Hall sets
If ж is a letter, then for any word w = hr ... hn, written as a decreasing
product of Hall words, the sequence s = (m, hlf ..., hn) is standard. Then
either m > and by (5.5.5) mw e M, or m < hr, and s -> (mhl, h2, • •., h„):
since m/ij < m by hypothesis, we have mhi eH\K and we deduce from
Theorem 4.3 (iii) and Lemma 5.1 l(i) that mw e M.
Now, let m be of length >2, and choose c = the greatest letter in m. If
c e H\K, then each letter of m is in H\K, hence m = m'b, m' e M, b e Л\К.
By the case |m| = 1, we have bw e M, hence mw = m'bw e M for any word w.
Suppose now that с e K. Let K' = {h e H | h > с}, M’ = {w e A* | Vk e K',
w < k}; then К' К, M' M, and by (i) and (ii), M' is a submonoid of A*,
generated by a suffix code X'; moreover, H\K' is a Hall set in the free
monoid X'* by (iii), K\K' is an upwards closed subset of H\K' and
M = {w e X'* | Vk e K\K', w < k}. Hence, M is obtained from X' in exactly
the same way as M from A.
Write m = hi... hn with h^H, hx >    > hn. Then by (5.5.5) we have
hi gH\K, hence ht gH\K'. Thus m e M' = X'*, by (5.5.5). The X'-length of
m is strictly less than its Л-length, otherwise each letter of m is in X' c M',
in particular c, a contradiction. Hence, by induction and the previous
remarks, we conclude that mp~1X'* M. Observe that, since c is the greatest
letter in m, m does not begin with c: otherwise its decreasing factorization
into Hall words begins by c, and m > c, hence m ф M‘, a contradiction. Hence
m = bm, b < c, thus b e M'. By the case |m| = 1, we have ЬА* M', hence
mA* M' = X'*. This shows that mpA* M.
If К Ф H, then Л/#{1} and the synchronizing property implies clearly
that X is complete.	□
Let X be a set of words of equal length n. Then evidently X is a suffix
code. We say that X is comma-free if whenever a word x of X is a factor of
a message (i.e. a word in X*), then the latter may be cut into two submessages
at x; formally, it means that
Vx e X, Vu, v e A*, uxv e X* => u,v e X*.	(5.5.6)
Note that if X contains two words x, y, with x = uv, у = vu for some
nonempty words u, v in A*, then X is not comma-free. Indeed, we have that
the word uvuv is in X* and contains the inner factor y, but u, v are not in
X*. This implies that the cardinality of X does not exceed the number of
primitive conjugation classes of length n. Hence
|.Y| < 1 EnW",
П djn
where q = |Л| (see Theorem 7.1). The next result shows that this bound may
be achieved, when n is odd.
Theorem 5.17 Let n be odd and q = |A|. Then there exists a comma-free code,
consisting of words of length n, and of cardinality (l/м) p(d)qnld.
5.5 Synchronous codes
123
The result is not true when n is even; see Berstel and Perrin (1985, p. 346).
To prove the theorem, we construct a special Hall set H in A, and show that
the set of words of length n in H is a comma-free code. This is enough,
because the number in the theorem is equal to the number of Hall words of
length n (Corollary 4.14).
Recall that M(A) denotes the free magma. We say that t g M(A) (respec-
tively A*) is even (respectively odd) if |t| is.
Lemma 5.18 There exists a Hall set H such that for any h, к in H:
(i) h even, к odd implies h < k;
(ii) h, к odd and \h\ > \k\ implies h < k.
Proof Let N = {t g M(A) | t even, or t odd and t" odd). Define a binary
relation z on N by s z t if either s is even and t odd, or if s, t have the
same parity and |s| > |t|. Then the reflexive and transitive closure of z is a
partial order on N. Extend it to a total order < on N. Note that for h, к
in N, one has (i) and (ii), and that h < Л" for each h in N. Define recursively
a subset H of N by A c H, and for any t = (f, t") in N\A, t is in H if and
only if t', t" g H,t' < t" and either t' g A, or (t')" > t". Note that h, к e N and
h < к implies (h, k) g N. Then H is a Hall set with the desired properties.
П
Proof of Theorem 5.17 Let H be the Hall set of Lemma 5.18. Let
К = {h g H | h odd). Then К is upwards closed in H, thus satisfies the
hypothesis of Theorem 5.16 and M = {w g A* | V/c g K, w < k} is a sub-
monoid of A*. Note that M is by (5.5.5) generated by the even Hall
words.
By Corollary 4.7, each word w in A* has a unique representation
w = kt . .. knm, with g K, m g M, кг >   • > k„. Moreover, if w is a right
factor of some word к in K, then by Lemma 5.9 we have w = kY ... k„ with
ktGH and kr >    > kn> k"; by (4.1.10) k" > k, so that each k, is in K,
because К is upwards closed.
We claim that each left factor of a word in H is in M u KM. Suppose the
claim is proved. Then suppose that the set of Hall words of length n is not
a comma-free code. This means by (5.5.6) that there are three such Hall
words h, к, I say, and a factorization hk = ulv for nonempty words u, v. Hence,
I = WiW2 with h = uWi and к = w2v. Since / is odd, one of and w2 is even,
and so there exists an even word w which is both right and left factor of
some word in K. Thus, by a previous remark, and the claim, we have
w = ki ... kn g M и KM, with kt g K, kt > • • • > kn. Since w is even, we must
have w g M, which contradicts the uniqueness of Corollary 4.7, in view of
(5.5.5).
124	5 Applications of Hall sets
To prove the claim, we prove first that for Hall words h, к one has
h, к odd, h < к => hk e M,	(5.5.7)
and
h even, к odd => hke KM.	(5.5.8)
Let us prove (5.5.7): if h e A or h" > k, then by (4.1.11), hk is a Hall word,
evidently even, so that hk e M. If h" < k, then, since h" is odd (because h" > h,
and К is upwards closed), we have by induction h"k g M; but h' is even, so
that h' g M, and hk = h'(h"k) g M, the latter being a submonoid.
We prove (5.5.8): we have h < к by Lemma 5.18(i). If h e A or h" > k, then
hk g H by (4.1.11), and hke К c KM. Otherwise h" < k: either h" is odd, so
that h' is too and, by (5.5.7), hk = h'(h"k) g KM; or h" is even, so that h' is
too, and by induction on |Л|, h"k = klm with кг g K, meM. Then, by
induction again, h'^ g KM, so that hk = h'h"к = h'^m g KMm 5= KM.
We now prove the claim: let w be a left factor of a Hall word h. If w is a
left factor of h', we are done by induction. Otherwise w = h'w', where w' is
a left factor of h". If w' = h", we are done because then w g H £ M и KM;
so we may suppose w' h". By induction, we have w' = m or km (m g M,
kGK). Thus w — h'm or h'km. If h' is even, w belongs to M и KMm, by
(5.5.8), hence to M и KM. If h' is odd, then, since h' < h" by (4.1.9), h" is
odd too; moreover, к is shorter than h" (because w' h”), so that by Lemma
5.18(ii), h" < k, hence h' < k; thus w g KM и M by (5.5.7). This proves the
claim.	□
5.6 APPENDIX
5.6.1 Long products of Lyndon words
There exists a function N(q, k) such that for any totally ordered alphabet A
with q elements, any word w in A* of length at least N(q, k) has a factorization
w = uli ... lkv, where each /t is a Lyndon word and f > •  • > lk (Reutenauer
1986a). For the proof, one introduces the code В = a(A\a)*, where a is the
smallest letter in A. Then В may be considered as a totally ordered alphabet.
The free monoid B* is embedded in A*, and one shows that: (i) the
alphabetical order in B* is the restriction to B* of that of A *; (ii) each Lyndon
word in B* is a Lyndon word in A*. Then the existence of N(q, k) is proved
by lexicographical induction on (k, q).
The previous result has as consequence a theorem of Shirshov, which has
itself applications in rings with polynomial identities (Shirshov 1957; see also
Lothaire 1983, Chapter 7). It has been extended to the Viennot factorizations
by Varrichio (1990). It is not known if a similar result holds for Hall words.
5.6 Appendix	125
5.6.2 Multilinear Lie polynomials
Let A = {a15 and call a polynomial P multilinear if it is a linear
combination of words aa(l)... aa(n), <reS„. Denote by Fn the space of
multilinear Lie polynomials. Then Fn is of dimension (n - 1)! and admits as
basis the set
[• • •	a<r(3)J’ • • • » a<r(n)l’ a 6 S„, cr( 1) = 1.
Indeed, an inductive use of Jacobi’s identity shows that these polynomials
generate F„. Moreover, they are linearly independent, because the above
polynomial is the only one involving the word ae(1)... aa(n). Alternatively,
one can use the fact that the number of multilinear Lyndon words is (n — 1)!.
Let A as above be naturally ordered, and consider the set of Lyndon words
as a Hall set, and let Pw denote the corresponding polynomial (as in Theorem
5.1). Then one has the following identity (Melancon and Reutenauer 1989):
1)   • flcr(n) *
<reSn
5.6.3 Another approach to Hall sets
Example 5.15 is directly related to the Lazard elimination process (see
Section 0.3). It allows a quite different approach to Hall sets. Let Я be a
Hall set in A*. If A has a greatest element z, then by Theorem 5.16(iii) the
set H n B* is a Hall set in B*, with В = (A\z)z*. Similarly, if (azn) denotes
the tree (.. .(a, z),..., z), then the homomorphism of magma h: M(B) -> M(A)
sending azn e В on to (azn) is injective, and the set Л-1(Я) is a Hall set in
M(B). This allows us to prove all the results on Hall trees and words.
Let us sketch for instance the proof of Corollary 4.4. Let w be a word in
A* and let z be the greatest letter in w. We may replace A by a finite alphabet
with greatest element z. Let В be as above. Then we may write w = znxr ... xk
with Xi e B. Then the word ... xk in B* is strictly shorter (as a word on
the alphabet B) than w, hence we conclude by induction that w has a
decreasing factorization into foliages of Hall trees. To prove uniqueness, one
notes that n is necessarily unique (it is the number of z’s at the beginning of
w, because each x in В begins by a letter distinct from z); then one uses
induction by passing to the word x'i ... xk.
To prove Theorem 4.9 one proves first that, with the previous notations,
one has the isomorphism of K-modules:
K[z](x)K<B> -+ K(A),
zn0xl ...xk^ znPXl... PXk(Xi e B),
where, for x = azr, Px is the Lie polynomial [.. .[[a, z], z],..., z], with r zs.
Compare this with Theorem 0.6.
126	5 Applications of Hall sets
The factorization A* = z*((A\z)z*)* is a special case of bisection. Viennot
(1978) gives similar isomorphisms in the case of general bisections (see also
Lothaire 1983, Proposition 5.3.11), and more generally, for left regular
factorizations of the free monoid. In particular, У (a, b) has a canoni-
cal decomposition as the (module) direct sum of the Lie subalgebras
^(re Q> + u oo), where is the space generated by the homogeneous Lie
polynomials P such that dega(P)/degfc(P) = r (see also Viennot 1974).
5.7 NOTES
Lyndon words appear in the work of Lyndon (1954, 1955a), and they are
used by Chen et al. (1958) to construct basic commutators of the free group;
Viennot (1978) and Lothaire (1983) also construct the Lyndon basis of the
free Lie algebra; an equivalent basis had been constructed by Shirshov (1958).
It was Viennot who showed that the Lyndon basis is a particular Hall basis,
once Hall sets have been properly generalized (see Section 4.5).
Theorem 5.3 is due to Schiitzenberger (1958), with improvements from
Melancon and Reutenauer (1989) and Melancon (1991). Note that condition
(iii) holds in any enveloping algebra, as the proof shows. Theorem 5.7 is
from Reutenauer (1990). The assertion on the length in Theorem 5.13(i) is
due to Viennot (1978), and Theorem 5.13(ii) was proved by Duval (1983) in
the case of Lyndon words. All other results of Section 5.4 are due to
Melancon (1992), who also proved that the order obtained in (5.4.1) in the
case of Lyndon words is the alphabetical order. Theorem 5.16 follows an
idea of Schiitzenberger (1958) and part (vi) is especially due to him (1986).
Theorem 5.17 is due to Eastmann (1965); the proof by Scholtz (1969) of this
result consists in constructing a special Hall set, so we have included it in
this book (see also the book on codes by Berstel and Perrin (1985, Theorem
5.3.8)). The problem of factorizing matrices of the form 1 — a — b — • • • leads
also to the construction of special Hall sets, by Good (1971), who calls them
standard lists.
6
Shuffle algebra and subwords
The shuffle algebra is a free commutative algebra over the set of Lyndon
words; this result is presented in Section 6.1, together with a precise identity
on the shuffle product of Lyndon words, which implies that actually there
is a canonical structure of algebra of divided powers. In Section 6.2 a
remarkable presentation of the shuffle algebra is given; the generators are
the nonempty words, and the relations the nontrivial shuffle products. In
Section 6.3 we introduce subword functions on the free group, the Magnus
transformation of the free group, the algebra structure on the module of
subword functions, and the fact that this algebra is generated by the
particular subword functions corresponding to Lyndon words. The main
tools are the concept of representative, or recognizable functions on the free
group, and the infiltration product of Chen, Fox and Lyndon. Section 6.4
presents the commutator calculus of P. Hall, and its generalizations. There
are many results involving the lower central series of the free group, the
Magnus transformation, and the algebra of subword functions.
6.1 THE FREE GENERATING SET OF LYNDON WORDS
Recall that the shuffle product ш was defined in Section 1.4, and that
with this product is a free commutative algebra (Corollary 5.5). We show
here that the set of Lyndon words is a free generating set of the shuffle algebra.
Let A be totally ordered and put on A* the alphabetical order. Recall that
a Lyndon word is a word w on A* such that Vu, v e A +, w = uv => vv < v
(see Section 5.1). This means that w is smaller than all its nontrivial proper
right factors.
Recall that each word w in A* has a unique decreasing factorization into
Lyndon words; this is a consequence of Theorem 5.1 (the set of Lyndon
words is a Hall set), and of Corollary 4.7 (each word has a unique decreasing
factorization into Hall words). We assume that К is a Q-algebra.
Theorem 6.1 (i) The shuffle algebra К (A) is freely generated by the Lyndon
words.
(ii) For each word w, written as a product of Lyndon words vv = l\ ... Г?
6 Shuffle algebra and subwords
128
(/j > • • • > lk; ir,... ,ik> 1), one has
--------— ш • •  ш l^ik = w + £ auu,
i i  •   i)c 	и < w
for some natural integers au.
Proof Note that (i) is an immediate consequence of (ii): by triangularity,
the polynomials
Qw = —^-~l^'in---inirk
q!... ikl
form a Z-linear basis of Z<4).
First note that Qw has coefficients in N, and that w appears in Qw with
coefficient a > 1. This is a consequence of the following general fact (which
may be verified directly from the definition of the shuffle product): if
u15..., uk are words and i15... ,ik are natural integers, then the coefficients
of the polynomial	are natural integers all divisible by
if.... ikl, and u'i ... uk appears in this polynomial.
Consider the set of Lyndon words as a Hall set (Theorem 5.1). This implies,
by Section 5.2, the existence of two dual bases, (Pw)weA- and (Sw)weA*, of
К<Л>. By Theorem 5.1, the basis (Pw) has the following triangularity
property: Pw = w + greater words. By duality, we deduce Sw = w + smaller
words.
Now, with w as in the statement of the theorem, we have by Theorem
5.3(iii)
s„=
tl!... ikl
Moreover, the same theorem shows that Sw has nonnegative coefficients.
Thus
Sw = w + sum of smaller words.	(6.1.1)
In particular, S) = / + other words, hence we obtain
SW = QW + QW	(6.1.2)
where Q'w has non-negative rational coefficients. Comparing (6.1.1) and
(6.1.2), we deduce that all the words appearing in Qw are <vv; moreover, w
appears in Qw with coefficient a > 1, hence we must have a = 1.	□
The following result will serve us later.
6.2 Presentation of the shuffle algebra	129
Corollary 6.2 A word w is a Lyndon word if and only if for each nontrivial
factorization w = xy, there exists a shuffle of x and у which is greater than w.
Proof If w is a Lyndon word and w = xy (x, у / 1), then by definition
w < у < yx, and we conclude because yx is a shuffle of x and y.
Conversely, suppose that w is not a Lyndon word. Then w is a decreasing
product of Lyndon words: w = l\'... If (f >    > lk; . . . , ik > 1). Since
w is not Lyndon, we have ц + • • • + ik > 2. Let w = xy, with x = f: this is
a nontrivial factorization of w. Then each shuffle и of x and у appears in the
polynomial If" ш • ш lff'k. By Theorem 6. l(iii), this implies that и < w.
6.2 PRESENTATION OF THE SHUFFLE ALGEBRA
Here, К is still a Q-algebra. Define, for each nonempty word w in A*, a
variable xw. Denote by X the set of these variables, and consider the algebra
of commutative polynomials K[X]. We have a linear mapping
ф: K(A) - K[X], wh+xJw/ 1), 1 t— 0.
Let J be the ideal of K[X] generated by the polynomials
ф(и ш r), u, v e A + .
We shall see that the shuffle algebra is isomorphic with K[X]//.
Actually, the isomorphism may be precisely described, by the use of the
logarithm.
As in Section 1.5, consider the complete tensor product
= к<л>® к<л>,
with the shuffle product at the left of ®, and the concatenation at the right.
Then, as in Section 3.2, consider the element log(£u6?1* и ® u); we saw there
that
i°g( E u(x)u) =Ew®n1(w),
\ueA* J »’
where лу is a degree-preserving linear endomorphism of Q<>4>, whose image
is in ®(.4) (Lemma 3.8). The adjoint endomorphism Ttf of is completely
defined by the equality
E ^(w) (x) w = log! X u®ul,	(6.2.1)
w	\иеЛ*	/
see (1.5.9).
130
6 Shuffle algebra and subwords
Theorem 6.3 Considering К (A) with its shuffle structure, let f:	->
be the K-algebra homomorphism defined by f(xw) = ft*(w), for any
nonempty word w. Then Ker(/) = J, f is surjective, and K(Aj ~ K[X]/.^.
Proof Let u, v be nonempty words. Then by Theorem 3.1(iv), we have
(ttfiw), и ш v) = 0 for any word w, because nfw) is a Lie element. Hence,
by duality (л*(и ш v), w) = 0, which shows that я*(и ш v) = 0. Observe that
= f ° ’A; hence, we deduce f ° i/z(u ш v) = 0, and У Ker f.
For the opposite inclusion, define L =	and let v:	-> L be
the canonical projection. We show that the series
£ H(w)w = £ v(xw)w,	(6.2.2)
weA*	weA*
is a Lie series in £<Л). Indeed, from Lemma 1.5, we have the identity
2 ° b(P) = r(P), for any polynomial P. We need the following facts: r(P) is a
Lie polynomial, <5 = (id (x) a) ° <5, a(l) = 1, b(P) =	(P, u ш v)u (x) v,
Л(и (x) v) = |u|ur, and all these mappings are if near, homogeneous, and
degree-preserving; see Section 1.3 and Proposition 1.8. The above identity
means that for any word w
£ (w, и ш г)|м| ua(v) = r(w).
U, V
In the sum, separate the term corresponding to v = l,u = w. Thus we obtain
|w|w = r(w)— £ (w, и ш r)|u|ua(v),
i
since the terms with u= 1 vanish because |1| =0. The series (6.2.2) is
therefore equal to
£ vo|/z(w)(|w|-1r(w) - |w|-1 £ (w, и ш r)|u|ua(v))
W^l	U, V # 1
= £ V°l//(w)\w\~ir(w) — T,
w# 1
where T is
T =	£	v°^(w)|w|-1(w, u ш r)|u|ua(r)
U, V, w # 1
= £ |u|ua(r)|iw|~ 1 v° I Yj (w,uinv)w
u,v # 1	\w # 1
because (w, u ш r) / 0 => |w| = |ur|. Now, the second summation is equal to
и ш v, so that by definition of./ and v, the series Tis equal to 0. This shows
that (6.2.2) is a Lie series. Hence, by Theorem 3.2(iii), its exponential is
6.3 Subword functions	131
defined by a shuffle homomorphism KfA) -> L, that is,
E v(xjw = log( X
w #1	\ue A* /
Applying the homomorphism fi ® id: и ® v —► fi(u)v,	-> LfA), we obtain
from (6.2.1)
!°g( E PMuj = £М(Ф =
\иеЛ*	/ w	к
because rcf(w) = f(xw).
So we deduce that v = fi° f, which implies that Ker f c Ker v =
It remains to show that f is surjective. We deduce from (6.2.1) and from
the definition of the product in j/
(_ l)*-i
f(xw) = nf(w) =	£	--------WjUJ---UJWt,
W = и 1 . . . Uk К
where the sum is over к > 1 and u, / 1. Hence,
(- l)fc ~1
f(xw) = w + E	,
к > 2	к
W-Ui . . .Uk
By induction on the length, each u, is in Im(/), hence so is their shuffle
product. Thus we deduce that w is in Im(/).	□
6.3 SUBWORD FUNCTIONS
A word и is a subword of a word w if и = ar ... an (n > 0, at g A) and if
w = гоа1Г1«2  • • vn-ia„v„,
(6.3.1)
for some words in A*. The binomial coefficient (") is defined as the number
of factorizations (6.3.1). Observe that if a is a letter, then
a"
ap
the ordinary binomial coefficient. A function A* -»• 7L of the form
w >—>
is called a subword function. Denoting by A* the characteristic series of A*,
i.e. A- = v. it is easy to verify that the subword functions are defined
132	6 Shuffle algebra and subwords
by the shuffle product
U LU A* = ). I Iw.
weA* \ W/
Similarly, a simple verification shows that, if w = a^ ... an (n > 0, a, e A),
then
(1 + «i)(l + a2)... (1 + a„) = £ Пи.
ueA* \ W/
We call Magnus transformation the homomorphism M from A* into the
multiplicative monoid	defined by
M(a) = 1 + a,
for any letter a in A. Then we have M(w) = £иеЛ» („)u for any word w in
A*. Actually, let F(A) denote the free group generated by A; it contains A*
as a submonoid. Since the series 1 + a are invertible in	the Magnus
transformation may be extended to a group homomorphism, still denoted
by M:
M: F(A) -> Z«4»,	a^\+a.
For an element g of the group F(A) and a word и in A*, we denote (®) the
coefficient of и in M(g). Thus
W)= Z	(6.3.2)
ueA* \W/
We still call subword function a function F(A) —► Z of the form g i—► (®).
We shall see that these functions have close connections with Lyndon
words and the free Lie algebra. We call space of subword functions the
subspace over Q spanned by all the subword functions on F(A). If a, are
two functions F(A) -> <Q>, their (pointwise) product is the function
ffl:F(A)^Q>, (ffl)(g) = a(g)f(g).
This is the way the Q-algebra of functions is defined on F(A). We suppose
that A is totally ordered and that A* gets the alphabetical order; Lyndon
words are defined in Section 5.1 (see also Section 6.1).
Theorem 6.4 The space of subword functions on the free group is a subalgebra
of the Q-algebra of functions on the free group. It is generated by the particular
subword functions
gi—+1 j, и Lyndon word.
\u/
6.3 Subword functions
133
Fig. 6.1
We shall see in the next section that the subword functions corresponding
to Lyndon words are actually a free generating set of this subalgebra
(Corollary 6.19).
In order to illustrate the theorem, consider the following example.
Example 6.5 Let a, b be two distinct letters and let w be a word in A*.
Then one has
w \Z w
abj \a
This relation is proved by seeing how a and ab are relatively located, as
subwords of w; there are four cases, shown in Fig. 6.1. The above relation
shows that the product of the two subword functions Q) and (*’), on the free
monoid, is in the space of subword functions. It also shows that (X) may be
expressed by a polynomial in the subword functions corresponding to the
Lyndon words a, ab, aab (we suppose a < b).
Let us call a function a: F(A) -> Q representative (or recognizable) if there
exists a finite dimensional vector space E over Q, a right action of F(X) on
E, a vector e in E and a linear function f on E such that for any g in F(/4)
«(0) = f(eg).	(6.3.3)
Lemma 6.6 (i) Representative functions form a subalgebra of the Q-algebra
of functions on F(A).
(i) If a representative function vanishes on A*, then it is the zero function.
(iii) Each subword function is representative.
134	6 Shuffle algebra and subwords
Proof (i) If a', a" are as in (6.3.3), then
(a' + «")(<?) = f(eg),
where £ = £'©£" (with action under F(A): (x1 + x”)g = x'g + x"g),
e = e' + e", and f(x' + x") = f(x') + f"(x"). Moreover, we have
(«'«" )(g) = f(eg),
with £ = £' (x) £" (with action (x' (x) x")g = (x'g) (x) (x"g)), e = ё (x) e", and
f(x ®x") = f(x')f"(x").
(ii) Each element g in F(A) may be written g = u0V! ги^ ... uk^1vk luk,
with к > 0, u,-, Vj g A*. We prove that a(g) = 0 by induction on k. If к = 0,
it is true by assumption. Let к > 1; let n be the dimension of £, where
£,/, e are as above. By the Cayley-Hamilton theorem applied to the
endomorphism x —► xv[ 1 of £, we have for some rational numbers rb..., r„
xv^n = r1xvi”+1 + r2xvi” + 2 + • • • + r„x,
for any vector x in £. With x = cuqv""1, this is
eu0Vi 1 = r^euQ + r2eu0v1 +  •  + r„euov"~1.
Multiplying on the right by иг . .. vk fik and taking the image under f, we
obtain
»(g) = n«(w0«i • • • Vk'uk) + r2ct(u0vlul ...	4) + •  •
+ '•na(uoi’i~1Hi • • • Л).
By induction on k, the right-hand side is 0. Thus x(g) = 0.
(iii) Let a(g) = (2) for some word и in A*. Let £ be the finite-dimensional
subvector space of	generated by the set P of words which are left
factors of u. Let v: 0<<Л>> -> E be defined by v(S) = £reP (S, v)v. Note that
v(ST) = v(v(S)T): indeed v(ST) = £reF (ST, v)v = Y»ep Yv = xy (S, x)(T, y)v.
Since xyeP=>xeP, this is equal to £rep Ei =xy(v(S), x)(T, y)v =
YveP(v(S)T,v)v = v(v(S)T).
Define a right action of F(A) on £ by the formula: Xg = v(XM(g))
for any X in £, g in F(A). This is indeed an action, because
(Xg)h = v(v(XM(g))M(h)) = v(XM(g)M(h)) = v(XM(gh)) = X(gh).
Let e = 1 e £, and f:E-> О, X н-+ (X, u). Then by (6.3.2), ct(g) = (M(g), u).
Since и is in P, this is equal to (v(M(g)), u) = (v(eM(g)), u) = /(eg). Hence,
a is a representative function.	□
Recall the notation w\I, for a word w of length n and a subset 1 of
[и] = {1,..., и} (see Section 1.4). Given p words u15..., up of respective
lengths ..., np, their infiltration product, denoted by ur |	| up, is the
polynomial
“i !• • 4 “p = E НЛ» • • •
6.3 Sub word functions	135
where the sum is extended over all n < nr ч----+ np and all p-tuples of
subsets of [n] such that [n] = (J i < j< p Л’ l;J = nJ for 7 = 1, • • •, n, and
where w = w(f,..., Ip) is defined by w\Ij = ujt for j = 1,..., p. The infiltra-
tion differs from the shuffle in that there may be overlappings between the
Uj when they appear as subwords of w (we do not require the Ij to be pairwise
disjoint). We call infiltration of ul,...,up a word appearing in their
infiltration product. Each shuffle of u15..., up is an infiltration, with the same
multiplicity, and each infiltration of u15..., up is either a shuffle, or of
length < luj + • • • + |up|. Examples:
ab | ac = abac + 2aabc + 2aacb + acab + abc + acb
= ab ш ac + abc + acb,
ab la = aba + 2aab + ab = abtna + ab.
Lemma 6.7 Let g, x, у be words in A*. Then one has
PYH = E СФ> w)(g\	(6.3.4)
\Xj\y/ weH*	\W/
Proof The formula is clear by inspection (compare with Example 6.5).
Proof of Theorem 6.4 Consider the function a: F(A) -► <□, defined by
, a / 9 \ 9
«(<?) =
WV.
E (xly,w)(g\
we A*	\W/
By Lemma 6.6(i) and (iii), it is a representative function. It vanishes on A*
by Lemma 6.7. So, by Lemma 6.6(ii), it is the zero function. In other words,
(6.3.4) holds for any element g of the free group F(/l).
This shows that the space of subword functions is closed under pointwise
product, and proves the first assertion of the theorem. We show now, by
induction, that each function (») is in the subalgebra M generated by the
functions (?), I Lyndon word. There is nothing to prove if w is a Lyndon
word, or if w = 1 (because (?) = 1). Suppose that w is not a Lyndon word.
Then, by Corollary 6.2, there exists a nontrivial factorization w = xy such
that each shuffle и of x and у is < w. Let к = (x ш у, w) > 0. Then, by the
first part of the proof,
PY^ = +
\Xj\y J	\W / ueA*	\^z
u # H'
Since each infiltration of x and y, is either a shuffle of x and y, or has a
length <|x| + |y| = |w|, we deduce that the summation is over words и with
either |и| = |w| and и < w (alphabetical order), or with |u| < |w|. By induction,
the corresponding functions (®) are in M', similarly, (?) and (?) are in M.
Hence, so is (®), because к # 0.
136
6 Shufflle algebra and subwords
6.4 THE LOWER CENTRAL SERIES OF THE FREE GROUP
If g, h are elements of any group, we denote as usual their commutator by
(g, h), that is
(g, h) = g~lh~lgh.
For future reference, we recall the easily verified identity
(f, gh) = (f, h)(f, g)((f, g), h).	(6.4.1)
Define the subgroups F„(.l) of F(A) recursively by FfA) = F(A), and
+ i(A) = subgroup generated by the elements (g, h), for some i, j with
g e Ft(A), h e FfA) and i + j > n + 1. This sequence of subgroups is called
the lower central series of the free group.
We now define special elements of F„(4). For this, take a Hall set H in
A* (see Section 4.1), and denote by the set of Hall words of length <n.
For h in H, define recursively the element (h) of the free group F(A) by
(Л) = h if h is a letter in A and, if h is of length >2, let h = h'h" be its
standard factorization. Then
(й) = ((hz), (h")).
Observe that if n = \h\, then (A)eF„: this is shown by a straightforward
induction.
Recall that the Magnus transformation
M\F(A) -> Z«4»,
and the subword functions F(A) -> Z have been defined in the previous
section.
Theorem 6.8 For each g in F(A), there exists a sequence (nh(g))helI of integers
such that for any N > 1
9= П W-'”modFN+1(/l),	(6.4.2)
where the product is taken in decreasing order. For any N, the exponents nh(g)
in this equality are unique. The functions nh: F(4) -> Z so defined form a free
generating set of the algebra of subword functions.
In order to prove the theorem, we introduce a binary relation depending
on the fixed integer N, on the set of sequences
s =	h?)	(6.4.3)
of Hall words of length <N and their inverses, which are standard; that is,
6.4 The lower central series of the free group	137
for each i, either ht is a letter, or /if > hi+15..., h„, where h{ = hfi" is the
standard factorization of /1, (see Section 4.1). Observe that if each /1, is a
letter, or if the sequence s is decreasing, that is, hx > • • • > hn, then it is
standard. Observe also that a subsequence of a standard sequence is
standard. A rise of (6.4.3) is an index i such that hi < hi+l. An inversion of
(6.4.3) is a couple (i,j) such that i < j and /i,- > /i . A legal rise is a rise i such
that
^i + i ^i + 2, • • •» Л„.	(6.4.4)
Note that these notions coincide with those introduced in Section 4.1 by
simply replacing (6.4.3) by (/i15..., hn).
Let the sequence s in (6.4.3) have the legal rise i. Then we replace the
elements hf1, hf+\ in s by another sequence, depending on the value of the
exponents, thereby obtaining a new sequence s'. We write
s -> s'
hence defining the binary relation To simplify, write Л,- = h, hi+1 = k. We
have several cases: the simplest one is when \hk\ > N; then
h±r, k±Y is replaced by k±r, h±Y.	(6.4.5)
For the remaining cases, we suppose |/i£| < N. Then
h, к is replaced by k, h, hk;	(6.4.6)
/i-1, к is replaced by k, (hk)~l, h~l.	(6.4.7)
For the remaining cases, let e (respectively o) be the greatest even (respec-
tively odd) integer such that \hke\ (respectively \hk° |) is < N. Then
h, k'1 is replaced by k~ \ h, hk2,..., hke, (hk°)~ \ ..., (hk)"1;	(6.4.8)
h~ \ k~1 is replaced by k"1, hk,..., hk°, (hke)~ \ ..., (hk2)"1, h"1. (6.4.9)
Lemma 6.9 (i) The sequence s' is standard.
(ii) There is no infinite chain s0 -> st	sn -► •• •
Proof We have h < k, because h, к is a rise of s. Moreover, if h is not in
A, then h" > k, because 5 is standard. Also, by (4.1.10), к” > к hence k" > h.
By (4.1.11), hk is a Hall word, and inductively, hkr is a Hall word for each
r > 1, (hkr)" = k, and hkr < к by (4.1.10). We use these facts without reference
in the sequel of this proof.
The sequence s is the concatenation of the three sequences u,(h±1,k±1)
and v; the sequence s' is the concatenation of u, x, and v, where x is one of
the sequences replacing (h±1,/c±1) and given by eqns (6.4.5>-(6.4.9). Note
138	6 Shuffle algebra and subwords
that if w is a word such that vv *1 appears in x, then
w = h, k, or hkr~, к > w and if w ф A, w" > k. (6.4.10)
(i) In order to show that s' is standard, we have to verify that if I, m are
two words in s' with l ф A and m at the right of I in s', then I" > m. This is
clear if I and m are in the sequences и or v, because s is standard. So we have
three cases:
(a)	I is in u, m is in x—then I" > к because s is standard, hence I" > m by
(6.4.10);
(b)	I, m are both in x—then Г > к > m by (6.4.10);
(с)	I is in x and m is in v—then к > m by (6.4.4) because h, к is a legal rise,
hence I" > m by (6.4.10).
(ii)	We may assume that the alphabet A is finite. Then H<N is finite. Let
E = {(h, k) | h,k &H<N,h < k}.
Then E is finite. Order E by (hr, kr) > (h2, k2) if either kY < k2 or kr = k2
and hr >degh2, where >degh2 means either > |й2| or l/ij = |й2| and
h1 > h2. Now order N£ lexicographically: then there is no infinite strictly
decreasing chain in N£. We show the existence of a mapping v from the set
of standard sequences into N£ such that s -> s' implies r(s) > v(s'). This will
prove (ii).
Define v(s\h k) to be the number of subsequences (h, k) in s; in other words,
it is the number of inversions (h, k) in s (note that h < к because (h, k)e E).
In view of eqns (6.4.5)-(6.4.9), we have v(s')(h k) = v(s)(hk) — 1. Suppose that
I < m: we show that for each inversion (/, m) in s', not already in s, we have
(/, m) > (h, k). This will imply r(s') < v(s). We have three cases.
(a)	I is in u, m is in x: then, since the inversion was not in s, we have by
(6.4.10) m = hkr, r > 1; thus m < к and (/, m) > (h, k).
(b)	I, m are both in x: then, since in each replacing sequence (6.4.5)-(6.4.9),
к appears only at the beginning, we have m k, hence m < к by
(6.4.10). Thus (/, m) > (h, k).
(с)	I is in x and m is in v: since h, к is a legal rise of s, we have by (6.4.4)
к > m. If m < k, then (/, m) > (h, k). Ifm = k, then I < k, hence by (6.4.10)
I = hkr, r > 0; moreover, r > 1, otherwise the inversion (/, m) is already
in s; hence, |/| > \h\, which implies I >deg/i and finally (/, m) > (h, k). □
Recall that in Section 4.2 we defined a homogeneous Lie polynomial Ph of
degree |/i| for each Hall word h, and that the family (Рй)йен forms a basis of
the free Lie algebra. If S is a series in	and P a polynomial in Z<X>,
we write
S = P + 0(Л"+1),
to express the fact that 5 — P is a series having no term of degree <n.
6.4 The lower central series of the free group	139
Lemma 6.10 (i) For a, b in A, one has
(1 + «)-1(l + b)-1(l +«)(1 + b)= 1 + £ (- l)i+W[«, b]. (6.4.11)
i,J>0
(ii)	For each Hall word h of length n, one has
M((h)) = 1 + P„ + 0(4"+1).
(iii)	If g 6 FN(A), then M(g) = 1 + 0(4N).
(iv)	For g e FN(A), let M(g) = 1 + P(g) + 0(4N+1). Then g^ P(g) is a
homomorphism from FN(A) into the additive group of homogeneous Lie
polynomials of degree N over Z.
Proof (i) The left-hand side of (6.4.11) is equal to
£ (- l)i+W )(1 + b + a + ab)
i,j>0	/
= E (- l)i+Jaibj + (-l)i+JalbJ+l
i,j>0	i,J>0
+ E (-1), + W«+ E (-l)(+7a'^ab.
i.J>0	i,J>0
The sum of the first two summations is equal to Ei>o ( — The third
summation may be rewritten E«>o (—	1 + Eu>o ( — )),+j+1a,bJ'ba.
Hence, the whole sum is
X ( —1)W + E (-l)‘«i+1+ E (-l)'+;+Wba
i > 0	i > 0	i,j>0
+ £ (—l),+7aibJab = 1 + E (~l)i+Jaibj(ab - ba),
i,j>0	i.J>0
which is as required.
(ii)	Let h = h'h" a Hall word of length n > 2, written in standard
factorization. Then, by induction, we have M((h')) = 1 + Ph + 0(4" + J) and
M((b")) = 1 + Ph" + 0(4" +1), where ri = \h'\, n" = \h"| and n = n' + n".
Then,
M((h)) = M(((h’), (h"))) = M((h'yl(h")-l(h')(h"))
= M((h’)) -1 M((h")) -1 M((h' ))M((h"))
= (1 + Ph. + 0(4"' + 1))-1(l + Ph. + 0(/1"'+1))-1
x (1 +Ph. + 0(4" +1))(l + Ph„ +0(4" +1)).
6 Shuffle algebra and subwords
140
By (6.4.11), this is equal to
1+ z (-1)‘+чра, + 0(лп'+1)йрй., + 0(л""+1)У
i,J>0
x [P„, + О(Л" +1), Ph„ + О(Л""+1)].
Observe that the term corresponding to i, j involves only words of length
> in' + jn" + n' + n”. Hence, for i or j > 1, this term is O(An + J). This implies
that
М((Л)) = 1 + [P„, + О(Л" +1), ph.. + О(Л"" + 1)] + О(Лп+1)
= 1 +[РЙ.Р^] + О(ЛП+1)
= 1 + Ph + О(Лп+1),
by definition of Ph (Section 4.2).
(iii)	This is clear for N = 1. For the general case, take g e Fh heFj
with i +j > N. Then, induction and eqn (6.4.11) show that M((g,h)) =
1 + О(ЛЛ). Moreover, if M(gr), M(g2) are both of the form 1 + О(ЛЛ). then
so is M(grg2 x). This proves (iii) by definition of FN.
(iv)	This is the consequence of a straightforward computation. □
Proof of Theorem 6.8 (i) Observe that if a standard sequence is not
decreasing, then it has a legal rise, e.g. the right-most rise. This implies by
Lemma 6.9 that each standard sequence may be rewritten, using the binary
relation into a decreasing sequence. In particular, this is the case for any
sequence of letters, with exponents +1. Now, define for the sequence s in
(6.4.3),
(s) = (h^...(hnr.
We show below that if s -> s', then
(s) = (s')mod FJV+1(X).	(6.4.12)
This will imply the existence of the expansion (6.4.2).
Eqn (6.4.12) is obvious by definition of FN+l when (6.4.5) is applied,
because
uv = vu(u, v),	(6.4.13)
hence (h)±l(k)±l = (k)±l(h)±l modFN+l. When (6.4.6) is applied, (6.4.12)
holds too, by (6.4.13). We also have
v(u, v)~ lu~1 = vv~ lu~ lvuu~1 = lv,
which implies that (6.4.12) holds when (6.4.7) is applied. Now, by (6.4.1),
1 = (u, vv~l) = (u, v~ x)(u, r)((u, v), v~l),
6.4 The lower central series of the free group
hence
141
(u,v !) = ((u, v), v '(u,v) \	(6.4.14)
Writing (uvn) for (.. .((u, v), v),..., v), we obtain from this identity, applied
to (uvn) and v instead of и and v,
((uvn), v~l) = ((iw"+1),	1)~1(iw"+1)-1.
Hence, by (6.4.14) again,
(u, r"1) = ((uv), v~l)~l(u, r)-1
= (uv2)(uv2, v~ l)(uv)~1
= (uv2)(uv3, v~i)~l(uv3)~ fuv)'1
= (uv2)(uv4')(uv4',v~1)(uv3)~1(uv)'1 =   
= (uv2)(uv4)... (uv2n )(uv2n, v ' ^(ur2" ~1) ~1 ... (uv3)'1 (uv) ~1.
(6.4.15)
Thus, by (6.4.13)
uv1 = v~ lu(uv2)(uv4)... (uv2n)(uv2n, v~1)(uv2n~1)~1 ... (uv3)~ l(uv)~1.
The latter identity shows that (6.4.12) still holds when (6.4.8) is applied. For
(6.4.9), one argues similarly, using the identity
u~lv~l = v~l(u, v'r)~ lu~l
= v~ l(uv)(uv3)... (uv2n~ 1)(uv2n, v~ J)~ l(uv2n)~ 1 . . .
x (uv4)~ l(uv2)~ lu~ \
by (6.4.15).
(ii) We prove at the same time uniqueness of the exponents nh(g), and the
fact that nh belongs to the space of subword functions. This will be done by
induction: assume the result is true for N — 1 (N > 1). We know that an
expansion (6.4.2) exists. By definition, FN+ i is contained in FN, so that (6.4.2)
implies
g = П mod fn-
By induction on N, we know that the exponents nh(g) are unique and that
the functions g h-+ nh(g) belong to the space of subword functions, for
\h\ < N — 1. Apply the Magnus transformation M to (6.4.2), using the
identity (Lemma 6.10(ii)):
M((h)) = 1+Ph+Th,
142	6 Shuffle algebra and subwords
where Th = О(Л|Л|+1). By Lemma 6.10(iii), the image under M of both
members of (6.4.2) coincide up to words of length N; let us express this with
the symbol =N:
М(й) =„ п *))" = п (1 + Л + г»)"'
heH.\	heH<H
= 111 ("‘\а + nr
heH.\ i>0 \ I /
(Hl \	(Hi. \
1	   ‘ (Л, + Л,)"    (П. +
ll	/	\ Ik /
where the second sum is over all hx > • • • > hk, Hall words of length <N,
and integers i15..., ik > 1.
Denote by M(g)N the homogeneous part of degree N of M(g). We obtain
M(g)H= E I ЕМ-M*- (64-16)
|*|=N	k>0 Vi /	\ h /
where the second sum is subject to the further condition that
l/ij,..., \hk\ < N, and where * is a polynomial of degree N depending solely
on ...,hk and ip...,/*.
We know by Theorem 4.9(i) that the polynomials Ph are linearly indepen-
dent. Fix h0 of length N; then there exists an homogeneous polynomial Q
of degree N such that (Pho, Q) = 1 and (Ph, Q) = 0 for any other h of length
n. Take the scalar product of Q with the last identity. We obtain that nho is
equal to (M(g), Q) plus a linear combination of products nhl ... nhj, with
\ht\ < N. Observe that in the previous computation, only M(g) and the
exponents nh depend on g: this proves uniqueness of the functions nh.
Moreover, by (6.3.2), induction, and Theorem 6.4, we deduce that nh belongs
to the space of subword functions.
(iii) Equations (6.4.16) and (6.3.2) show that the subword functions belong
to the algebra generated by the functions nh. Hence, the latter functions
generate the algebra of subword functions. They generate it freely, because
for any finite subset H' с: H, one can find, by (6.4.2), an element g in F(X)
such that nh(g) takes, for h in H’, arbitrary values in Z.	□
Remark 6.11 In practical computations, it is useful to add to eqns (6.4.5)-
(6.4.9) the rule
h, h"1 or Л"1, h is deleted in s.	(6.4.17)
Indeed, Lemma 6.9 still holds, as is easily verified (for (ii) one has to add to
the vector v(s) the length of s as a new component, at the extreme right).
Moreover, it is clear that rule (6.4.17) does not change (s), with the notations
of the proof of Theorem 6.8.
6.4 The lower central series of the free group
143
Example 6.12 Take the Hall set of Example 4.6, g = bab1. and N = 4. We
only need to know the inequalities b > ab2 > a2b2 > ab3 > ab > a > a2b.
Then, by using at each step the right-most rise, we have
Hence, we have
(b, a, b"1) -► (b, b~l, a, ab2, (ab3)~ \ (ah)~l)	by (6.4.8)
-> (a, ab2, (ab3)"1, (ab)"1)	by (6.4.17)
-> (ab2, a, a2b2, (ab3)-1, (ab)-1)	by (6.4.6)
-> (ab2, a2b2, (ab3)- \ a, (ab)-1)	by (6.4.5)
-> (ab2, a2b2, (ab3) - \ (ab)"1, a, (a2b)~1	) by (6.4.8)
bab 1 = (ab2)(a2b2)(ab3) fab) га(а2Ь) ^odFj.
Corollary 6.13 (We use the notation of Theorem 6.8.) There exist nonnegative
integers kh u (b e H, ueA*, 1 < |u| < |b|) such that for any g in F(A)
и \U/
(6.4.18)
Proof In view of Theorem 6.8 and Lemma 6.6 it is enough to prove (6.4.18)
when g is a word in A*. Observe that when dealing with standard sequences
in A*, the only rule of the rewriting system -» which is used is
, . .	... fk,h,(hk) if\hk\<N,
h, к is replaced by<
И }[k,h if \hk\>N.
We shall use a modified version of this rewriting system, which works on
labelled standard sequences, i.e. sequences of the form
S = ((hl,El),...,(hn,En)),	(6.4.19)
where s = (hx,..., hn) is a standard sequence and each £, is a subset of M.
If hh hi+l is a legal rise of s, then we define
5' = (..., (/!,•_!,£,•_!), (hi+l,Ei+1),(hi, Ei),(hihi+l, £,• u £1 + 1),
(hi + 2,Ei + 2),...),	(6.4.20)
where the term with A,b1+1 has to be omitted if this word has length >N.
Then S' is still a labelled standard sequence, and we write 5 -» S'. It is easy
to verify that -> is confluent, because legal rises do not overlap (cf. proof of
Theorem 4.3(i)). Then one shows that the reflexive and transitive closure
of -> is also confluent (cf. proof of Theorem 4.3(i)) and that there is no
infinite chain So Sj S3... (cf. Lemma 6.9(ii)). Thus, for any S, there is
144	6 Shuffle algebra and subwords
a unique final S', i.e. a sequence such that S S' and that S' -> S" for no
sequence S". We write S' = f(S). Now, let и = ar ... an e A+ (a; e A, n > 1)
and
5 = ((fli, (a2, {i2}),..., (a„, {i„})),
where i15..., i„ are distinct numbers.
With f(S) given by (6.4.19), let
kh,u = |{i | 1 < i < n, ht = h and £f = {i\,..., in}}|.
Observe that khu is well defined, i.e. does not depend on the sequence
i15..., in of distinct numbers. Observe also that by definition of f(S) and of
we have |£,-| < |/i,|, so that kh u # 0 implies |u| = n < |/i|. We show that
(6.4.18) holds.
For a sequence 5 as in (6.4.19) and £ с N, define S|£ to be the sequence
obtained by keeping only those i with £; c £. We claim that if 5 T, then
S|£ T\E. Indeed, we may suppose 5 -► S’, that hf, hi+l is a legal rise of
5 and that S' is given by (6.4.20). Then either £,• and Ei+1 are both contained
in £, so that £,• u £i+ j is too, hence S\E -> S’\E; or one of £, or £i+1 is not
contained in £, so that neither is £;u£i+1, and S|£ = S’\E. From the
claim, we deduce that for any word w = a^ ... an, Ec{l,2,...,n] and
S = ((«i, {1})’ • • • ’ («„’ {"}))’ we have
/(S|£)=/(S)|£,
because the underlying standard sequence of f(S) is decreasing, hence so is
that of/(S)|£. Recall the notation w\E, defined in Section 1.4. Then we have
nh(w) = number of (h, £) in f(S), with E {1,..., n}
= E E number of (h, E) in f(S)\E
ueA* w|£ = u
= E E number of (h, E) in f(S\E)
ueA* w|£ =u
= £ Z Z	□
ueA* w|£=u	ueA* \H/
Corollary 6.14 (i) For geF(A) and heH, the number nh(gn) is a linear
combination over Z of ("), 1 < i < \h\.
(ii) For gx, g2 6 F(A), and heH, the number п^д^^1) is a polynomial over
Q in the numbers n^g^, nh2(g2), /ц, h2 e H.
Proof (i) We have, by Corollary 6.13,
nh(g”)= E kh,u(g\
1 < |u| < |h| \U /
6.4 The lower central series of the free group	145
Let M(g) = 1 + T with (T, 1) = 0. Then M(g") = Ei>o (")T‘ and
(0"') = (M(9").“)= 1 ("V',U)= E f"V',u),
\U /	i>O\l/	0<i<|u|M/
hence the result follows.
(ii) This has a similar proof.
Corollary 6.15 An element g in F(A) is in Fn(A) if and only if M(g) =
1 +0(Л"). In this case, nh(g) — 0 for any Hall word h with \h\ < n — 1, and
M(g) = 1 + P + 0(Л"+ 1)/or some homogeneous Lie polynomial P of degree n.
Proof The direct part follows from Lemma 6.10(iii). Conversely,
suppose that M(g) = 1 + 0(Л"). By Theorem 6.8, we have
g = П|я| < л-i (h)nh{e) mod F„(X). Let |h| < n — 1: by Corollary 6.13, we have
nh(g) = Ei <|U| <:«-1 fch,u(u) = °, by hypothesis. Hence g = 1 mod. Fn(A). The
last assertion follows from Lemma 6.10(iv).	□
Corollary 6.16 The group Fn + 1(A) is contained in F„(A), and F„(A)/Fn + 1(A)
is an abelian group, freely generated by the classes of the elements (h), for h
a Hall word of length n. This group is canonically isomorphic with the ^.-module
of homogeneous Lie polynomials of degree n. The dual group (Fn/Fn+l)* is
freely generated by the functions nh, for h a Hall word of length n.
Proof That Fn+ 1(Л) c Fn(A) is an immediate consequence of our definition
of Fn. If g,he Fn(A), then (g, h) g Fn+ fA), hence gh = hg mod Fn+ fA), which
shows that Fn(A)/Fn+ fA) is abelian. Each element g in F„(X) has by Theorem
6.8 and Corollary 6.15 a unique expansion
в = П (M"h<9)modFn+1,
|Л| ="
which shows that the classes of the (h) (respectively the nh) such that \h\ = n
freely generate Fn/Fn + l (respectively its dual group).
The assertion about Lie polynomials follows from Lemma 6.10(ii), (iv):
the mapping g P(g) defines an isomorphism between the two /-modules,
because the basis ((h)) of Fn/Fn+ j is mapped on to the basis (Ph) of the space
of homogeneous Lie polynomials of degree n.	□
Take now as a Hall set the set L of Lyndon words.
Corollary 6.17 The dual group (Fn(A)/Fn+l(A))* is freely generated by the
subword functions (9, I a Lyndon word of length n.
Proof By the previous proof, Fn/Fn+l is isomorphic with <fn, the /-module
146
6 Shuffle algebra and subwords
of homogeneous Lie polynomials of degree n, via the mapping g P(g) of
Lemma 6.10(iv) with N = n. Now, the dual group of Tn is freely
generated by the linear functions P h-+ a(, where a, is the coefficient of P when
expressed in the basis (P(), / a Lyndon word of length n. Since Pt = l± greater
words (Theorem 5.1), we deduce by triangularity that JSf* is also generated
by the functions P (P, /). To conclude, we observe that (M(g), I) = (?), by
definition of the subword functions.	□
Corollary 6.18 For any finite set L' of Lyndon words and any sequence
(ai)ieL' °f integers, there exists an element g in F(A) such that
9
I
= Otl,
VI eL’.
Proof Let n be the maximum length of the elements in L. By induction on
n, there exists an element gx in F(A) such that: (?) = az for any / in L of
length <n. By Corollary 6.17, there exists an element g2 of F„(4) such that
(?) = — (?) f°r any I in L' of length n. By Corollary 6.15, we have
M(g2) = 1 + 0(4"). Let g = gvg2- Then for / in L
= (W), /) = (M(9l)M(g2), I)
= E (M(^i)’ w)(W2), v)
I — uv
= (М(д,'),Г)+ £ (M(91),u)(M(92),i.).
I — uv
М>л
If \l\ < n - 1, the second sum is empty and (?) = (М(^/), /) = (»/) = az. If
\l\ = n, then the second sum reduces to (M(g2), I), hence (?) = (?) + (9Z2) = az.
□
Corollary 6.19 The subword functions on F(A) (respectively on A*) cor-
responding to Lyndon words are algebraically independent in the algebra of
functions over F(A) (respectively 4*).
Proof Let P(xt) be a polynomial in the commuting variables xh I g L.
Define Ф(д) = P(ff)) for any g in F(A). We have to show that Vw g 4*,
<D(w) = 0 => P = 0.
Now, by Lemma 6.6, Ф is a representative function on F(4), vanishing on
4*, so Ф(д) = 0 for any g in F(4). Let L' be the finite set of Lyndon words
such that %,, I g L', appears in P, and let (az)Zgr be any sequence of integers.
Then, take g as in Corollary 6.18; hence, 0 = Ф(д) = P(az). This implies that
P vanishes for any integer choice of the variables. Hence, P = 0.	□
6.5 Appendix
147
Corollary 6.20 Suppose that A is finite. A series S in 2«Л» with constant
term 1 is in the A-adic closure of M(F(A)) if and only if for any words x, у
in A*, one has
(S, x)(S, y)= £ (x j y, w)(S, w).
we A*
Proof The conditions are necessary by Lemma 6.7. Conversely, let S satisfy
these conditions. Let n > 0. Then we may find g e F(A) such that (?) = (S, I),
for any Lyndon word / of length < n (Corollary 6.18). An argument similar
to that used in the proof of Theorem 6.4 shows that, in fact, we then have
(5, w) = (®) for any word w of length <n. Hence, S and M(g) coincide up
to words of length n. This shows that 5 is the limit of a sequence of elements
in М(Р(Л)).	□
6.5 APPENDIX
6.5.1 Lie polynomial basis of the shuffle algebra
Let L be any basis of the free Lie algebra over Q. Then L is a free generating
set of the shuffle algebra 0<Л> (Perrin and Viennot 1981). We may assume
that the alphabet is finite. Then the space of polynomials of degree < n is of
finite dimension; since the scalar product (, ) is positive definite, Theorem
3.1(iv) implies that each polynomial P of degree n, with (P, 1) = 0, may be
written
P = Q + £ * u ш r,
where Q is a Lie polynomial and u, v are nonempty words with |u| + |v| < n.
By induction on n, we conclude that P may be expressed as a linear
combination of shuffle monomials in the elements of L. A counting argument,
using the Witt formula (Corollary 4.14), shows that these monomials are
linearly independent.
Let Vn denote the subspace of 0<Л> generated by the shuffle products of
n Lie polynomials. The previous result shows that there is a direct sum
а<л> = ф v„.
л>0
6.5.2 A shuffle subalgebra
Let L be a subset of Л* which contains each left factor of each word in Ц
then the shuffle subalgebra of О<Л> generated by L is a free commutative
algebra, over some subset of L. The main step to this result is the proof of
the following lemma: if V c A* and ueA* are such that V u {u} contains
148	6 Shuffle algebra and subwords
all its left factors, and if и is algebraically dependent on V in the shuffle
algebra, then и is in the shuffle subalgebra generated by V. Hint: apply the
derivation Qi-'+Qa-1 of the shuffle algebra (cf. (1.4.3)) to an algebraic
dependence relation of и on V, for some letter a, and use induction.
A particular case of the previous result is when L is the set of factors of
a given word w = a{ ... an (af g A). Define for each letter a a square matrix
<pa of order n + 1 by ((ра){ i+l = 1 if a{ = a, and (cpa)i j = 0 otherwise. The
dimension of the Lie algebra generated by the matrices <pa is equal to the
transcendence degree of the shuffle algebra generated by the factors of w
(Reutenauer 1985b).
6.5.3 Homomorphism
Let Ml be an algebra over K, and p: KfAy -> End(M) a homomorphism
from the concatenation algebra into the algebra of linear endomorphisms of
M. Suppose that ffla) is a derivation of M for any letter a. Let (p: M -> К
be an algebra homomorphism. Define a mapping p: M -> К ((A}} by
p(m) = J (p(p(w)(m))w.
weA*
Then p is a homomorphism of K-algebras, from M into the shuffle algebra
Kf(A')') (Fliess 1981, Proposition III.l).
A particular case of this result is when A has only one letter a. Then the
shuffle algebra К<Л> is isomorphic with the algebra К [[a]], via the
isomorphism £n>0 an«" £n>o (ап/и!)а" is supposed to contain Q).
Then, if A is a linear differential operator of K[x15..., x„] and a15..., a„
elements of K, the mapping
K[xi,...,x„] ->	zH->exp(aA)z|Xi = ai,
is a homomorphism of K-algebras (see Grobner 1967, pp. 16 17).
6.5.4 Causal analytic functionals
Let A = {«j,..., am}, and (ui)l^i^m a family of piecewise continuous
functions [0, T] -> (R. For each word w and t e [0, T], define the iterated
integral J'o dw recursively by J'o dw = 1 if w is the empty word and, if w = uah
then J'o dw = J'o (Jo du)u;(s) ds. Note that this definition agrees with the
definition in Section 3.1 when a,(t) = J'o u,(s)ds.
Let 5 g	such that |(S, w)| < C|w|!r|w| for some constants C and r.
Define
y(t)= X I dvv’	(6.5.1)
weA* Jo
6.5 Appendix	149
which is a convergent series. Then the functional (ub ..., u„) у is called
a causal analytic functional, with generating series S. The product of two such
functionals corresponds to the shuffle product of their generating series (Fliess
1981). The proof is similar to that of Corollary 3.5, using integration by
parts.
Among these functionals, there is the special class of those which
correspond to a differential system of the form
m	I
4(0 =	E	ихолм,
• = i	?	(6.5.2)
y(t) =	h(q),	J
where q(t) belongs to an analytic variety Q over R, and where the vector
fields Л15..., Am, and the function h: Q -> R are analytic in a neighbour-
hood of q(0). The corresponding generating series 5 is of finite Lie rank,
i.e. the vector space {/’ 5 | P g 5T(A)} is finite dimensional, where
P S = Е»<=л* (S, wP)w, i.e. 5 h-+ PS is the adjoint of the right multiplica-
tion by P (Fliess 1983). The finiteness of the Lie rank is equivalent to the
following condition: 5 belongs to a finitely generated shuffle subalgebra of
R«4», closed under the operations Tt—>T°P (Pg (R<X>) and closed in
the Л-adic topology (Reutenauer 1985«).
6.5.5 Differential algebra
A special case of system (6.5.2) is the case where Q is a finite-dimensional
vector space, and At,h are linear. Such a system is called bilinear in control
theory. It corresponds to the case where the generating series S is recogniz-
able (see Section 1.6.8).
Let A = {u15..., am} and R{u} = R{uH..., um} the (R-algebra of differen-
tial polynomials, i.e. the algebra of (commutative) polynomials in the
variables ul,...,um and their formal derivatives u\, u'{,.... Consider the
algebra M = R«/l»® R{u}, with the shuffle structure on R<</1>>. It is
isomorphic with (R{u})<4>, with its shuffle structure, which is isomorphic
with an algebra of formal power series in (infinitely many if m > 2) commuta-
tive variables, by Theorem 6. l(i). In particular, M is without zero divisors,
and we may form its field of fraction K.
The algebra M (hence the field K) becomes a differential ring if one defines
as derivation the unique derivation D extending that of R{u}, and which is
defined on R<<4>) by
m
D(S) = E (Sarl)®uh	(6.5.3)
i= 1
where Sa-1 = Е»<=л* (S, wa)w. Observe that eqn (6.5.3) is motivated by the
150	6 Shuffle algebra and subwords
functional intepretation (6.5.1): there, the derivative of у is given by
m /	(*t	\
y(t) = X ( E (s>wai) dwjUi(t).
i = 1 \weA*	Jo	/
The differential field К has no constants others than R, i.e. if a g К and
D(a) = 0, then a e R. This may be shown by proving first a lemma in the
shuffle algebra, interesting in itself: if Sa~1 ш T = 5 ш Ta~1 for any letters
a, then 5, T differ by a multiplicative constant.
The result on constants in К allows to apply the Picard-Vessiot theory
of linear differential equations (see Kaplansky 1957). It is shown in Fliess
and Reutenauer (1983) that recognizable series satisfy a linear differential
equation with coefficients in R{u}, which has all its solutions in K.
Furthermore, the Galois group of the splitting field of this equation, and its
Lie algebra, are characterized via the syntactic algebra of the series.
* For the differential algebra approach to control theory, see Fliess (1989).
6.5.6 Subword order
Write и < v if и is a sub word of v: this is the subword order. A theorem of
Higman asserts that if the alphabet is finite, then each set of pairwise
incomparable words is finite (see Lothaire 1983, Theorem 6.1.2). The Mobius
function of this order has been computed by Bjorner (1989); it uses a variant
of the binomial coefficient („) of Section 6.3. Define („)„ to be the number
of factorizations (6.3.1) such that: Vi = 0,..., n, \/aeA, A*aaA* and
г(^я,Л*. Then the Mobius function fflu, w) is equal to (— l)|u| + |w|(u)„. In
other words, for u / w, one has E« < <> < » F(v-> w) = ® = Ци<»<» F(u-> r)-
shown in Bjorner and Reutenauer (1992) that the series
(w \
j и ® w
It / n
is rational.
6.5.7 Recognizable subsets of Г(Л)
A subset L of F(A) is recognizable, or representative, if for some finite group
G and some homomorphism <p: F(A) -> G, one has L = (p~v(p(L). In this
case, we say that L is recognized by G. The family of subsets of F(A) which
are recognized by finite p-groups (respectively nilpotent groups) is equal to
the boolean algebra generated by the particular subsets
g e F(^)
I = i mod p
и/
(u g Л*, i > 0)
151
6.5 Appendix
(respectively the particular subsets
g g F(/l)
g \	,
J = i mod n
и/
(u g Л*, n > 1, i > 0).
This may be shown by using the Magnus transformation, considered with
coefficients in Z/pZ (respectively Z/nZ), and mod O(Ad+ 1) for suitable d, to
show one inclusion. For the other, use Theorem 6.8, Corollary 6.13, and the
following result, proved similarly to Lemma 6.6: the function
is an N-linear combination of subword function (“'). This result allows one
to reduce the knowledge of n = („) mod pk to that of
n \	,
. mod p,
.P'J
because
n \	л
. | = и,- mod p,
P'J
if n = £ n.p1 is the p-adic expansion of n (the latter congruence is obtained
by expanding (1 + x)" in characteristic p).
6.5.8 Quotients of the lower central series and free Lie algebra
Let (F„)„>i be the lower central series of the free group F over A and
consider the set gr(F) = £„> i Fn/Fn+ P Then gr(F) becomes a natural
structure of graded Lie algebra over Z. Indeed, FnFn.{ is an abelian group,
hence a Z-module. Now, let x e F„/F„+1, у e Fp/Fp+15 respectively represented
by f e Fn, g e Fp. Then (/, g) e Fn + p and formula (6.4.1) shows that the class
mod Fn + p+ j of (/, g) depends only on the class of g mod Fp+l, i.e. on y. A
symmetric argument shows that (/, g) depends only on x. Hence we have a
well-defined mapping
(Fn/Fn +1) x (FP/FP +1)	Fn + P/Fn + P+^	(x, у) н-» [x, y].
This mapping is Z-linear, in view of (6.4.1). Extend this mapping to gr(F)
by linearity. Since (/,/) = 1 and (g,f) = (f,gY\ we have [x. x] = 0 for
any x in gr(F). Now, the Jacobi identity is a consequence of the following
identity, where fe denotes g~\fg:
152	6 Shuffle algebra and subwords
From this, one may deduce that the mapping g i—»P(g) of Lemma 6.10(iv)
induces an isomorphism from gr(F) onto ^(Л), and give another proof
of Corollary 6.15. In particular, gr(F) is the free Lie algebra over Z; see Serre
(1965).
This method is actually the original proof of Witt (1937); see also Lazard
(1954) and Bourbaki (1972).
6.5.9 Image of the Magnus transformation on A*
The following result (Ochsenschlager 1981; see also Lothaire 1983, Theorem
6.3.22), characterizes the image of the restriction of the Magnus transforma-
tion to A*: a polynomial P in М<Л> is in M(A*) if and only if for any words
x, у one has
(P, x)(P, y) = X (* 1 b H’)(p’ w)-
weA*
In contrast to Corollary 6.20, no closure is needed here.
6.6 NOTES
Theorem 6.1 is due to Radford (1979), who proved combinatorially the
triangular identity in the statement. The first assertion was also proved by
Perrin and Viennot (1981). The proof given here follows Melancon and
Reutenauer (1989). It is not true in general that any Hall set freely generates
the shuffle algebra. Corollary 6.2 is due to Chen et al. (1958). Theorem 6.3
and its proof are due to Ree (1958); note the unusual role played by the
logarithm in this proof (as Ree observes). We have borrowed the terminology
‘subword function’ and ‘binomial coefficient’ from Eilenberg (1976); the
extension of this terminology to the free group is done via the Magnus
transformation. Another approach is to take the free differential calculus of
Fox (1953). Theorem 6.4 is due to Chen et al. (1958); they introduced the
infiltration product and proved Lemma 6.7. Lemma 6.6 follows Melancon
and Reutenauer (1993). See Chapter VI of Lothaire (1983) for more on
subwords.
The commutator calculus of Section 6.4 has its origin in a paper of
P. Hall (1933); he essentially proved eqn (6.4.2) of Theorem 6.8, by the use of
his ‘collecting process’; this process generates particular Hall sets: those
where the order is compatible with the length (cf. the discussion in Section
4.5). See also P. Hall (1957) and M. Hall (1950, 1958, Chapter 11), where
uniqueness of the exponents in Theorem 6.8 is also proved, for these
particular Hall sets.
Here, we work with the general Hall sets, as generalized by Viennot (see
Section 4.5); the corresponding group commutator calculus was developed
6.6 Notes
153
by Melancon (1991), and Melancon and Reutenauer (1993); Gorchakov
(1969) already gives some of the results (see also Ward 1969). The algorithm
presented here is a generalization of the collecting process of P. Hall,
with ideas of M. Hall (1959), Schiitzenberger (1958), and Melancon
and Reutenauer (1989) where standard sequences of Lyndon words are
introduced.
Lemma 6.10 is due to Magnus (1937). Corollary 6.13 is due to Therien
(1983), in the case of the particular Hall sets (see above) and when g is a
word in A*; its generalization to general Hall sets, and to the free group
follows Melancon and Reutenauer (1993). Corollary 6.14(i), which is an
immediate consequence of Corollary 6.13, is related to an identity of P. Hall
(1933; see also Magnus et al. 1976, Theorem 5.13B). Corollary 6.15 is due
to Magnus (1937) and Witt (1937); they also proved the existence of the
canonical isomorphism between Fn/Fn+l and the Z-module of homogeneous
Lie polynomials of degree n (Corollary 6.16). Corollaries 6.17-6.20 are all
due to Chen et al. (1958).
For applications to the Burnside problem and the theory of p-groups, see
Magnus et al. (1976, Chapter 5) and M. Hall (1959, Chapters 12, 18).
7
Circular words
There are many links between the free Lie algebra and circular words; the
most immediate is the equality of the homogeneous dimension of the former,
given by the Witt formula, and of the number of primitive necklaces. The
aim of this chapter is to study circular words as an end in itself.
We begin by computing the number of primitive necklaces, and of
necklaces. Then, we describe the bijection between primitive necklaces and
Hall words. In the next two sections two efficient algorithms are described;
the first generates Lyndon words up to a given length, and the other
computes the factorization into Lyndon words of a given word. The
decreasing factorization of a word into Hall words provides a bijection
between words and multisets of primitive necklaces; this bijection depends
of course on the chosen Hall set. In the final section we give another bijection,
which leaves invariant the associated permutation, and which has applica-
tions in the study of the various symmetric functions related to the free Lie
algebra.
7.1 THE NUMBER OF PRIMITIVE NECKLACES
We say that two words u, v in A* are conjugate if for some words x, y, one
has и — xy and v = yx. The relation ‘u and v are conjugate’ is an equivalence
relation, called conjugation. An equivalence class is called a conjugacy class,
a circular word, or a necklace. Geometrically speaking, a necklace is a regular
и-gon in an oriented plane whose vertices are labelled in Л; two such n-gons
are considered as identical if they may be superposed by applying a rotation,
a translation and a homothety (see Fig. 7.1).
A necklace is called primitive if no nontrivial rotation leaves it invariant;
a word is called primitive if its conjugacy class is a primitive necklace. More
generally, a necklace has always a smallest period d, dividing its length n,
and in this case the corresponding conjugacy class has d elements, and is
of the form C = {u”/d,..., unJd}, where {tq,..., ud} is a primitive conjugacy
class. We say in this case that each word w in C has period d and exponent
n/d (see Fig. 7.2).
7.1 The number of primitive necklaces
155
ababb
babba
abbab
bbaha
babab
Fig. 7.1 The conjugacy class of ababb.
Theorem 7.1 Let A = {a1?..., aq} be an alphabet with q elements. The
number of primitive necklaces of length n is
1 E Kd)qn/d.
П d\n
The number of primitive necklaces having щ occurrences of the letter a^
(i = 1,..., q) is
(7.1.1)
(7.1.2)
1 У (d)[ nld
nd|^, \njd,..., nq/d) ’
with n = щ + • • • + nq.
Here, p and ( ) stand, respectively, for the Mobius function and the
multinomial coefficient.
156
7 Circular words
Theorem 7.1 may be proved directly, but we prefer use the formalism of
generating functions, which will also be useful in the sequel. Let xb ..., xq
be commutative variables. The evaluation of a necklace и is the monomial
x"1 ... xq4 in Q[x1?..., xj if и has occurrences of the letter ah for
i = 1,..., q. The evaluation of a word is defined similarly. If L is a set of
necklaces (respectively of words), then the generating function of L is the sum
of the evaluations of the elements of L. For example, the generating function
of the set of primitive necklaces of length 2 is
Z
i<J
because each such primitive necklace has a unique representative of the form
0,0^, i < j. Define the nth power sum pn to be the symmetric polynomial
p„(xi,...,xe) = x'; + --- + xj.
Theorem 7.2 The generating function of the primitive necklaces of length n is
1 X pWpT-	(7.1.3)
П d | n
Proof Let Ce(n) denote the set of words of length и and period e. We have a
partition of the set An of words of length и
A" = U
e| n
Let P(e) denote the set of primitive words of length e. The mapping
и н» unle, P(e) - Ce(n),
is a bijection. Observe that each word in P(e) has e conjugates, each with
the same evaluation, and that er(u"/e) = er(u)|Xi_x?,e, where ev denotes
the evaluation. Moreover, the generating function of An is clearly
(Xj + • • • + xq)n = p"(xi, • • •, xq). Thus, denoting by /e(x1?..., xq) the gener-
ating function of the primitive necklaces of length e, we have
rt(x„ ..., x,) = £ ele(x"C,  • , <')•	<71-4>
e | n
Note that pd(xl,..., xq) = pfx*,   , xq). So we have
- E	• • • ,xq) = - £ n(d)p"l,d(xdl, ...,xd)
n d | n	П d | л
= 1 X M(d) X ................xj"),
П d | л e| n/d
by (7.1.4) with n replaced by n/d and x, by xf.
7.1 The number of primitive necklaces	157
Since d\n and e\n/d is equivalent to e\n and d\n/e, this is equal to
- E e/e(x"/e’---’x«/e) E /*00-
71 e | n	d | n/e
The second sum vanishes, unless n = e in which case it is 1, so that the whole
expression is equal to l„(xb ..., xq).	□
Proof of Theorem 7.1 In formula (7.1.3), put xb ...,x = 1; this gives
formula (7.1.1). Moreover, the number of primitive necklaces with n,
occurrences of the letter at (i = 1,..., q) is the coeffcient of x"1 ... xnqq in
(7.1.3); this coefficient is (7.1.2), because
p^Xj,...,^) Г E f W’.-.x^’.	□
ri + • • • + rq = n/d vp • •  » rqJ
Corollary 7.3 The generating function of the necklaces of length n is
- E <p(d)Pdd,
П d\n
where (p is the Euler function. The number of necklaces of length n on an
alphabet with q letters is
1 E 4>(d)qn/d.
П d\n
The number of necklaces having П; occurrences of the letter ahi = 1,..., q, is
1 E
П d | л,
/ n/d
Xnjd,..., nq/d.
Proof For each conjugacy class C of words of length n, there is a unique
conjugacy class C of primitive words of length dividing n, n/e say,
such that C =	| и e C'}. This implies the following identity between the
generating functions kn (respectively /„) of necklaces (respectively of primitive
necklaces):
k„(x1,...,x,)= E U(xi’• • • ’-x«)-
е|л
By Theorem 7.2, we deduce
158
7 Circular words
kn(xl,...,xq)=Y-iT E
е|л П/e f\n/e
= E - E
e | n П ef\n
= 1 E pnAxi’- • ->*q) E
ft d\n	e\d \ej
= - E
ft d [п
because p/xl,..., xq) = pe/(Xi,..., xq) and <p(d) = E e\d ep(d/ef as is well
known.
In order to obtain the other two formulas, we follow the proof of Theorem
7.1.	□
7.2 HALL WORDS AND PRIMITIVE NECKLACES
Conjugacy of words is an equivalence relation, which preserves primitivity
and periodicity (see Section 7.1). Observe that if a set H of words is a set of
representatives of the primitive conjugacy classes, then the set
{hn | heH,n > 1}
is a set of representatives of all the conjugacy classes of positive length.
The following result is a particular case of a theorem of Schiitzenberger
(1965).
Theorem 7.4 Let H be a subset of A*, with a total order <. Suppose that
each word in A* has a unique factorization
Iii .. .hn,n>0,hiE H,hx > •  • > hn.	(7.2.1)
Then H is a set of representatives of the primitive conjugacy classes.
Recall that Hall sets are defined in Section 4.1. Then the previous theorem
and Corollary 4.7 immediately imply the next result.
Corollary 7.5 Let H be a Hall set in A*. Then each word in H is primitive,
and each nonempty word is conjugate to a unique word of the form hn, he H,
n > 1.
For the proof of Theorem 7.4, observe that a series of the form
[5, Г] = ST — TS is a linear combination (possibly infinite) of polynomials
uv — vu, u, v e A*.
7.2 Hall words and primitive necklaces	159
Proof of Theorem 7.4 Let X = {xh | h g H} be an alphabet in bijection with
H. Then, an easy extension of the Baker-Campbell-Hausdorff formula
(Corollary 3.4) shows that the series
log I f ] eXh I
\heH J
is a Lie series. It is easy to see that its component of degree 1 is Ehew xh.
Thus, the series
log (fl eXh)- E xh	(7.2.2)
\heH / heH
is a Lie series without component of degree 1, and is therefore an infinite
linear combination of polynomials of the form uv — vu, u, ve X*.
Since each word in A* has a unique factorization (7.2.1), we have in
О«Л»
(l - Л)-1 = П (1 - hr1,
heH
where we write A for Еаел a. Taking logarithms, we obtain
£-'/!" = logfnd-*)’1)-	(7.2.3)
n > 1 n	\heH	/
Consider the (continuous) homomorphism f:	-*
xh i—> log( 1 — h)-1. Under this homomorphism, (7.2.2) becomes
!og( П 0 “	~ E log(1 -hy1,
\heH	/ heH
which is therefore an infinite linear combination of polynomials of the form
uv — vu, u, v e A*. By (7.2.3), we deduce that so is the series
E -An- e E -hm-	(7.2.4)
n> 1 n	heH m> 1 m
Let C be any conjugacy class of length n. Taking scalar product of E«ec u
with (7.2.4), we find that ^|C| = E^ec m- If C is primitive, |C| = n and the
left-hand side equals 1; by primitivity, the right-hand side is Ehec b hence
there is exactly one h in H n C. If C is not primitive, |C| < n, hence H n C
is empty.	□
Let H be a fixed Hall set in A*. There is a simple algorithm which gives
for each word vv / 1 the unique hn (heH,n> 1) conjugate to w. This
algorithm works on sequence of Hall words, takes as input w considered as
160
7 Circular words
the sequence of its letters, and produces as output the sequence (h,..., h) of
h repeated n times, such that he H, n > 1 and w is conjugate to hn.
As in Section 4.1, let h = h'h" denote the standard factorization of a word
he H\A. We say that a sequence s = (ftp ..., h„) of Hall words is cyclically
standard if for any i=l,...,n, either h, is in A, or h, = h\h" with
h'- > hp ..., hn. Observe that a sequence of letters is always cyclically
standard. Moreover, if the h, in s are not all equal, then there is an index i
such that
h,-<hi+i =sup{hp...,h„},
where the indices are taken modulo n. For such an i, replace s by s with
_ J(hp...,hi_ph1hi+phi+2, ...,h„) if i < n;
((h„hph2,. ,.,h„_1)	if i = n.
Observe that either h, is a letter, or hf = h\h", in which case h'- > hi+l because
s is cyclically standard: in both cases, h = hthi+ , is a Hall word with h' = hf,
h" = hi+p by (4.1.11). Moreover, s is cyclically standard, because on one
hand h" = hi+l > hp ..., hn (by assumption on i) and h" > h by (4.1.10),
and on the other, s being cyclically standard, we have h" > hk, for any j, k,
and in particular hj > hi+x = h" > h.
For s as above, let w(s) = h{ ...hn. Then clearly, w(s) and w(s) are
conjugate words. The algorithm consists to apply the transformation s -» s
repeatedly, starting from a sequence of letters, until a sequence of the form
(h,..., h) is obtained.
Example 7.6 Consider the Hall set of Example 4.6; we want to find the
Hall word conjugate to the word abaahaba. Knowing that aabab, aab, a, ab, b
are Hall words, written in increasing order, the algorithm gives:
(a, b, a, a, b, a, b, a) -* (ab, a, a, b, a, b, a) -» (ab, a, ab, a, b, a) -» (ab, a, ab, ab, a)
-» (aab, a, ab, ab) -* (aab, aab, ab) -» (aab, aabab)
-» (aababaab).
The latter word is the desired Hall word.
Let the alphabet A be totally ordered, and order A* alphabetically (see
Section 5.1). Recall that a Lyndon word is a word which is smaller than all
its nontrivial proper right factors.
Corollary 7.7 A word w is a Lyndon word if and only if it is primitive and if
it is the smallest word in its conjugacy class.
Proof The set L of Lyndon words is a particular Hall set (Theorem 5.1).
Hence, by Corollary 7.5, L is a set of representatives of the primitive
7.3 Generation of Lyndon words	161
conjugacy classes. Now, if w is a Lyndon word, let vu be a conjugate of w,
with w = uv, u, v / 1. Then, by definition of a Lyndon word, we have w < v.
Since v is shorter than w, w is not a prefix of v, so that by Lemma 5.2, we
deduce w < vu. Hence, each Lyndon word is the smallest element of its
conjugacy class, which concludes the proof.	□
7.3 GENERATION OF LYNDON WORDS
Let A be a totally ordered finite alphabet, let A* be ordered with the
corresponding alphabetical order, and L be the corresponding set of Lyndon
words. We fix an integer N > 1, and denote by LN the set of Lyndon words
of length < N. Define a function
In- Ln\{z} -* Ln,
(where z is the greatest letter in A) by the condition: lN(u) is the smallest
word, for the alphabetical order, in the finite set {x e LN \ x > u}.
Observe that lN is simply the ‘next element function’, in the finite totally
ordered set LN (whose greatest element is z). Hence, the knowledge of lN will
provide a simple algorithm for the generation of all Lyndon words of
length <N: starting by a (the smallest letter), compute и = lN(a), then lN(uf
and so on, until z is obtained.
The next result, due to Duval (1988), shows that the function lN may be
computed very efficiently. For this, we need two definitions. Given a word
u, call periodic expansion at length N of и the word DN(u) = ukp, where к and
p are uniquely defined by к e N, p is a nonempty prefix of u, |t/p| = N. In
other words, DN(u) is the unique prefix of length N of the ‘infinite word’
uuu .. .u ... Note that such a prefix, being of length N > 1, may always be
written as ukp, with p a nonempty prefix of u.
Now, define a function 5: A*\z* -* A*, where z* denotes the set of powers
of z, by
5(w) = min{x g A* | x > w, |x| < |w|}.
There is an easy algorithm to compute 5(w). Let w = vaz', where a is a letter
distinct from z. Let b be the letter following a in the finite totally ordered
set A. Then
S(w) = vb.	(7.3.1)
Theorem 7.8 The function lN is equal to the composition S DN.
We first prove several lemmas.
162	7 Circular words
Lemma 7.9 If u, v are Lyndon words such that и < v and n, p are integers > 1,
then unvp is a Lyndon word.
Proof The case n = p = 1 is eqn. (5.1.2). The general case follows by
induction, once it is observed that и < unvp < v (the latter inequality because
unvp is a Lyndon word).	□
The next lemma shows that there is no Lyndon word between a Lyndon
word and its periodic expansion at any order. It is a first step towards the
proof of Theorem 7.8.
Lemma 7.10 Let u, w be Lyndon words with |u| < N, and и < w < DN(u).
Then w = u.
Proof Suppose that w satisfies и < w < DN(u). We show that w has a
nontrivial suffix s and и has a prefix p such that s < p. Hence, s < p < и < w
and w is not a Lyndon word.
First, suppose that w is a prefix of DN(u). Then, since DN(u) is a prefix of
some power u“ of u, we deduce that w is also; hence, w = ukp for some prefix
p of и and some integer к > 0; we have w 1 and we may choose p to be
nontrivial (replacing if necessary p = 1 by p = и and к by к — 1). Hence,
p = s is a nontrivial suffix of w = ukp.
Suppose now that w is not a prefix of DN(u). Since w DN(u), we have by
definition of the alphabetical order, w = xaw', DN(u) = xbt for some words
x, w’, t and letters a < b. We have, by definition of DN, x = uku', with к 0
and p = u'b prefix of u. Let s = u'aw'; then w = uku'aw' = uks, and s is a
nontrivial suffix of w. Moreover, s = u'aw' < u'b = p.	□
Lemma 7.11 Let и = ps be a Lyndon word, s 1. Let c be a letter such that
s < c. Then pc is a Lyndon word.
Proof Let t be a nonempty proper suffix of pc. We show that pc < t, which
will prove that pc is a Lyndon word. We have t = t'c, where t' is a proper
suffix of p. Then t’s is a nonempty proper suffix of ps, and therefore ps < t's,
because ps = и is a Lyndon word. Moreover s < c, hence t's < t'c, and p is
a proper prefix of ps, hence p < ps. Thus p < ps < t's < t'c = t, and p < t.
Suppose that p is a prefix of t; since t is a proper suffix of pc, we have
| p| + 1 = |pc| > |t| => | p| > t|, and we would have p = t, which is not true
because of the previous inequality p < t. Hence, we can use Lemma 5.2(i) to
deduce that pc < t.	□
Corollary 7.12 Let и be a Lyndon word, which is not the greatest letter of
the alphabet, p a nonempty prefix of u, and к an integer. Then S(ukp) is a Lyndon
word.
7.4 Factorization into Lyndon words	163
Proof Let z be the greatest letter in A. Then p does not begin with z: indeed,
otherwise и = zu' and и > z > last letter of u; since и is a Lyndon word, this
implies that и is equal to its last letter, hence и = z, against the assumption.
In particular, p is not a power of z. Hence, we have p = p}azl. where a g A\z,
and by (7.3.1), S(p) = pffi, where b is the letter after a in A. We have и = p^
for some word Sj beginning with a. Hence, < b, and by Lemma 7.11,
we have that pffi is a Lyndon word. Now, by (7.3.1) again, we have
S(ukp) = S (икр{аг1) = tZpjb. Since pta is a prefix of u, we have и < ргЬ, and
we conclude by using Lemma 7.9.	□
Proof of Theorem 7.8 Let и be a Lyndon word of length <N, и / z, and
w = lN(u). Then by definition w is the smallest Lyndon word in the set
(x g A* \ x > u, |x| < N}.
By Lemma 7.10, we have w > DN(u). Hence, w is the smallest Lyndon word
in the set {x e A* | x > DN(u), |x| < N}. But S(DN(u)) is the smallest word in
this set. Since DN(u) = for some integer к and some nonempty prefix p
of u, we know by Corollary 7.12 that 5(DN(u)) is a Lyndon word. Hence,
S(Djv(u)) = w = lN(u).	□
Example 7.13 N = 9, A = {a < b}. Then и = aabbb is a Lyndon word. One
has /9(u) = S D9(u) = Sfaabbbaabb) = aabbbab.
7.4 FACTORIZATION INTO LYNDON WORDS
As in the previous section, L denotes the set of Lyndon words relative to
the alphabetical order on A*, where A is a totally ordered finite alphabet.
By Theorem 5.1, L is a particular Hall set, hence, by Corollary 4.7, each
word w in A* has a unique decreasing factorization.
w = f...ln,	(7.4.1)
Existence and uniqueness of the factorization (7.4.1) may be proved
directly—indeed, each word has a factorization into Lyndon words (e.g. w
is the product of its letters). Now, take such a factorization, with a minimal
number of factors. Since for к, I Lyndon words, к < I implies kl g L (Lemma
7.9), this minimal factorization must be decreasing. This proves the existence
of factorization (7.4.1). Uniqueness is a consequence of the following
result.
Lemma 7.14 For a factorization of the form (4.1) of the word w, the following
properties hold:
(i)	ln is the smallest nontrivial suffix of w;
(ii)	ln is the longest suffix of w which is a Lyndon word;
(iii)	li is the longest prefix of w which is a Lyndon word.
164	7 Circular words
Proof Let s be a nontrivial suffix of w. Then s = //, + j ... where /• is a
nonempty suffix of /, and 1 < i < n.
(i)	Ц is a Lyndon word, hence we have /, < < Г{11+ { . ln = s, and /„ is
the smallest nontrivial suffix of w, because l„<
(ii)	Suppose that s is longer than Then i < n, which implies /'• < s.
Arguing as in (i), we deduce ln < s, which shows that s has a nontrivial suffix
smaller than itself, and is therefore not a Lyndon word.
(iii)	Let p be a prefix of w, strictly longer than /P Then p =	. lj-il'j,
where l'j is a nonempty prefix of lj and 2 < j < n. Using (7.4.1), we deduce
l'j < Ij <	... lj-il'j = p, which shows that p is not a Lyndon word. □
In order to find the Lyndon factorization of a word, one may apply the
algorithm described in the proof of Corollary 4.4. However, there is a much
more efficient algorithm due to Duval (1978, 1983), which we describe now.
For this purpose, call sesquipower of a word и any word of the form ukp, for
some integer к > 0 and some prefix p of u, which we may assume to be
proper (i.e. p / u), without loss of generality. Such a sesquipower is called
nontrivial if к > 1. Denote by 5 the set of nontrivial sesquipowers of Lyndon
words. An element и of 5 has always a representation
и = lkp, I e L, p proper prefix of I, к > 1.	(7.4.2)
For a given element и of S, the representation (7.4.2) is unique: indeed, let
p =	... hq be the decreasing factorization of p into Lyndon words; then
hi < p < I, hence the decreasing factorization of и into Lyndon words is
lkhi ... hq, and we conclude by uniqueness of this factorization.
In the sequel of this section, it should be understood that when we deal
with elements of S, we deal actually with their unique representation (7.4.2).
For a word и in 5, having the representation (7.4.2), we may write I = pas
for some letter a and some word s. We denote a = g(w). Now, define a binary
relation on the set 5 x A*, denoted by and defined for any и in S, b in
A, and v in A* by
(u, bv) -» (ub, i?) if s(u) < b.
This is well defined, because e(u) < b implies ub g 5: indeed, let и = lkp as in
(7.4.2) and I = pas, thus a = e(u). Then either a = b, hence ub = (pas)kpa is
clearly in S; or a < b and as < b, hence by lemma 7.11, pb is a Lyndon word
with I < pb, which implies by lemma 7.9 that ub = lkpb is a Lyndon word,
hence in 5.
Denote by the transitive closure of This is clearly a partial order
on 5 x A*, and we say that xeS x A* is maximal if for no у in 5 x A*, one
has x -* y. It is clear that for each x in 5 x A*, there is a unique maximal
у in 5 x A* such that x y.
The factorization algorithm is described in the next theorem. Note that
1А Factorization into Lyndon words	165
the decreasing factorization into Lyndon words of a word w may be written
w = lkl...lk?, Ц >   • > lp, kl?..., kp>	(1A3)
Theorem 7.15 Let w be a word, factorized as in (7 A3), c its first letter, with
w = cw', (u, v) the unique maximal element in S x A* (where и is as in (7.4.2)),
such that (c, w')^ (u, v). Then l{ = I and = k.
In other words, the rewriting system -» allows us to compute the power
of the first Lyndon word in the factorization of w; then, one continues with
pv instead of w, and so on.
Example 7.16 w = abbabbababb, a < b. For each и in S, written as in (7.4.2),
we put the letter e(u) in bold face. Then we have:
(a, bbabbababb) -» (ab, babbababb) -» (abb, abbababb) -* ((abb)a,bbababb)
-» ((abb)ab, bababb) -» ((abb)2, ababb) -» ((abb)2a, babb)
-» ((abb)2ab, abb).
The latter is maximal, because b > a. So w = (abb)2s, and we continue with
s: (a, babb) -* (ab, abb) -* ((ab)a, bb) -» ((ab)2, b) -» (ababb, 1). Hence, s is a
Lyndon word, and the factorization of w is (abb)2(ababb).
Proof Since (c, w')^ (u, v) we have w = cw' = uv = lkpv. Let I = pas, ae A.
Let pv = hi ... hq be the decreasing factorization into Lyndon words of pv.
Then either hi is a prefix of p, hence h{ < p < I, or p is a proper prefix of
hi: pbh', where b is the first letter of v; then, since (u, v) is maximal, we must
have a > b, hence hi = pbh' < pas = /. In both cases, hi < I, which shows
that the decreasing factorization into Lyndon words of w is /% ... hq and
that lk = lkl, hi...hq = lk22... If.	□
The proof shows that this algorithm is linear in time: more precisely, to
factorize w, one needs at most 2|w| comparisons between letters in A.
7.4.1 Applications
(a)	We know by Corollary 7.5 that each primitive word is conjugate to a
unique Lyndon word. To find it, it is enough to factorize ww into Lyndon
words, and to extract from this factorization a Lyndon word of length |w|.
Indeed, such a word will clearly be conjugate to w. Moreover, it exists: indeed,
let I be the unique Lyndon word conjugate to w. Then w = xy, ух = I. Hence,
ww = xly. If x or у is empty, we are done because ww = //. So we may suppose
that x, у / 1. The last factor in the Lyndon factorization of x is a suffix of
x, hence of /, so greater than /. The first factor in the Lyndon factorization
of у is a prefix of y, hence of I, so smaller than /. Thus, the Lyndon
166
7 Circular words
factorization of xly is obtained by concatenating that of x, I, and that of y:
hence, I appears in the Lyndon factorization of ww = xly.
(b)	Each Lyndon word I of length >2 has a standard factorization I = IT'
(see Section 4.1), which according to the proof of Theorem 5.1 is given by:
I" is the smallest nontrivial proper suffix of /. In order to find Г, I", let I = aw,
ae A, and let w = f ... ln be the Lyndon factorization of w. Then I" = ln.
This is an immediate consequence of Lemma 7.14(i).
Hence, one can quickly compute the standard factorization of any Lyndon
word w, and by iterating this process, its associated tree t(w) (cf. the proof
of Theorem 5.1).
7.5 WORDS AND MULTISETS OF PRIMITIVE NECKLACES
Recall that, for a set E, a multiset of elements of E is a mapping M: E -» N.
For e in E, M(e) is the multiplicity of e in M. It is finite if its cardinality, i.e.
£ee£M(e), is <oo.
Given a subset H of A* which satisfies the hypothesis of Theorem 7.4,
there is an evident bijection, given by (7.2.1), between the words of A* and
multisets of elements of H. This is true for example for any Hall set H, or
for the set H = L of Lyndon words (cf. Corollary 4.7 and Theorem 5.1).
Moreover, Corollary 7.5 shows that there is a bijection between H and the
set of primitive necklaces. Hence, we have the following result, where the
evaluation (respectively length) of a multiset is the product (respectively sum)
of the evaluations (respectively lengths) of its components, with multiplicities.
Theorem 7.17 Given a Hall set on A* (especially the set of Lyndon words),
there is a canonical evaluation-preserving bijection between the three following
sets.
(i)	The set of words of length n.
(ii)	The set of multisets of length n of Hall words.
(iii)	The set of multisets of length n of primitive necklaces.
We give now another bijection between words and multisets of primitive
necklaces, which has better invariance properties than the previous ones.
This bijection will be useful in the study of the various symmetric functions
related to the free Lie algebra.
Let A be a totally ordered alphabet. Let w = at ... an in A* (a; e A). Let
[n] = {1,..., n] and define a function dw: [n] i-> A x [n] by d(i) = (ah i).
Evidently, bw is injective. Order A x [n] with lexicographic order. Then the
condition dw(i) < &WU) defines a total order on [и]. Note that this condition
is equivalent to
(«,<«;) or (a( = a}and i <j).	(7.5.1)
This total order on [n] is called the standard numbering of w: it consists of
7.5 Words and multisets of primitive necklaces	167
numbering the positions of the letters in w, from left to right, starting with
the smallest letter in w, continuing with the second smallest, and so on (see
Example 7.18).
The standard permutation of w is the unique permutation st(w) = a of [n]
such that a(i) < a(j) is equivalent to bw(i) < bw(j). One obtains a directly,
viewed as a word on [n], if one numbers the positions of the letters in w, as
above.
We leave the verification of the following fact to the reader:
a(i) = |{j | 1 < j < n, as < a{}| + |{j | 1 < j < i, at = a7}|.
Example 7.18
1	2	3	4	5
w = b	b	a	a	b
a = 3	4	1	2	5
6	7	8	9	10	11	12
d	d	d	b	d	b	c
9	10	11	6	12	7	8
With w and a as before, consider a cycle c = (f,, ik) of a. Then denote
by vf(w) the necklace which is the conjugation class of the word ahai2... aik.
Then define a multiset of necklaces M(w) to be the multiset {vf(w) | c cycle
of st(vv)}. More formally, M(w) is defined for any necklace v by M(w)(v) =
number of cycles c of st(w) such that vf(w) = v.
Observe that the evaluation of w is equal to the evaluation of M(w).
Example 7.19 With w and a as in Example 7.18, the cycles of a are shown
in Fig. 7.3. The multiset M(w) is obtained by replacing each digit i by a„ i(i)
(instead of a{—this does not change the necklaces); see Fig. 7.4.
Fig. 7.4
168
7 Circular words
Theorem 7.20 The mapping w i—> M(w) is an evaluation-preserving bijection
from the set of words onto the set of finite multisets of primitive necklaces.
The inverse bijection W is described in the proof.
Proof (a) We construct below an evaluation-preserving mapping W from
the set of finite multisets of primitive necklaces into A* such that M ° W = id.
Then W will be injective, hence surjective by Theorem 7.17, because the set
of words with a given evaluation is finite. Hence, M is a bijection from A*
onto the set of multisets of primitive necklaces.
(b)	Let P be the set of periodic sequences in ЛЧ Each element in
P is of the form л(и), where n: A+ -* P is the mapping n(b1 ... bk) =
(b1,...,bk, bx,..., bk,...), (bi-e A). Observe that л defines a bijection
between primitive words and periodic sequences. Let z: Л141 -* Л141 be the shift
mapping, defined by z(a0, ar, a2,...) = (ar,a2, a3,...). Its restriction to P is
a permutation of P. One has
z(n(b1 ... bk)) = Ti(b2 ... bkbf),	(7.5.2)
and
z~1(ri(bl ... bk)) = ittb^ ... Ьк_г).	(7.5.3)
This shows that the orbit under z-1 of л(и), и primitive, consists in the
sequences л(г), v conjugate to u, and has precisely |u| elements, the number
of conjugates of u.
Put on the lexicographic order, and do the same for Л141 x N. Let
f\ Л 4 -♦ A be the mapping sending each sequence on its first element. Extend
z and f to P x N by z(s, /) = (z(s), /) and f(s, I) = f(s).
(c)	Denote by (u) the conjugation class of a word и and by u the reversal
of и (defined by u = ak... аг if и = ax ... ak, a,-e A). Let N be a finite multiset
of primitive necklaces of length n. We associate with N the subset £ of P x N
defined by
E = IJ {л(й)} x [(V((u))],	(7.5.4)
ueA +
where [0] = 0. Note that E is of cardinality n, because there are |u| elements
in the conjugation class of a primitive word u. Note also that E inherits the
total order < of Л4 x N; we write E = [ег < e2 < • • • < en}. If и and v are
conjugate words, then Ar(('<)) = N((v)) and й and v are conjugate. Hence, by
(7.5.2) and (7.5.3), the restriction of z to E is a permutation E -* E. Let a be
the permutation of [n] defined by a =	1 ° z~ 1 ° b, where b(i) = e{. Observe
that <5, b~1 are increasing functions. Let w = at ... an, with a{ = f °z~
We show that M(w) = N, which will finish the proof: it suffices to pose
w = W(N).
(d)	We claim that a is the standard permutation of w. Indeed, o(i) < a(j)
is equivalent to z~1(ei) < z~1(ej) (because <5, <5-1 are increasing),
7.5 Words and multisets of primitive necklaces	169
which by the properties of lexicographic order, is in turn equivalent to
{/(^-1(e,)) </(z-1(e7))} or {/(z-1(e,)) =/(z-1(e;)) and e{ < ef, that is,
(a, < af) or (a, = aj and i < j), which is (7.5.1). So the claim is proved.
By (7.5.3) and (7.5.4), a cycle of z1 is of the form
(л(Ьк... bj, /), (Ti(blbk... b2), I),..., (n(bk_1 ... bf>k), I),
for some primitive word и = b{ ... bk and some I, 1 < I < M((u)). Note that
conjugate words determine the same cycle. Since a = d1 °z1 ° <5, the cycles
of a are of the form с = (ц,..., ik) with
*i =	...bt),Z)), i2 = b'1((Ti(bl...b2),l)),...,ik
= d~1((Tt(bk_l...bk),l)).
Note that a, = f °z-1(e,.) = f °z~x °<5(0, hence by (7.5.3)
a,. = fo l((it(bk ... bf>, I)) = f^b. ...), 0) = b.,
ai2 = f°z~1((-n(b1 ... b2), /)) = /((л(/>2 ...),/)) = b2,
aik = foZ-\(n(bk^ ... bk), I)) = f((n(bk ...), /)) = bk.
This shows that the corresponding necklace vf(w) = (ait ... aik) is equal to
(u), and we conclude that M(w) = M.	□
Example 7.21 The inverse mapping W of M is described on the multiset
of primitive necklaces shown in Fig. 7.4. Label each vertex of each necklace
by the periodic sequence obtained by reading, in the opposite direction, the
necklace (see Fig. 7.5). Now, order these sequences lexicographically (when
multiplicities occur, one has to order the multiple necklaces first; see Fig.
7.6). One obtains a permutation a in cycle form; write it as a word, and
replace each digit i by the label of the corresponding vertex:
12 3 4
<7 = 3	4	1	2
w = b	b	a	a
5	6	7	8
5	9	10	11
b	d	d	d
9	10	11	12
6	12	7	8
b	d	b	c
One obtains the word of Example 7.18, as expected.
Fig. 7.5
170
7 Circular words
Fig. 7.6
7.6 APPENDIX
7.6.1 Lyndon elements in free partially commutative monoids
We use the same notation as Section 4.4.2. Let A be totally ordered, consider
the alphabetical order < on A*, and order M = M(A, 6) in the following
way: denote by ip: A* -* M(A, 0) the canonical morphism and let
st(m) = max(<p~ ^m)) for any m in M; then m < p if st(m) < st(p).
An element m of M is called a Lyndon element if for any nontrivial
factorization m = pq, one has m < q. The properties of Lyndon elements in
M are quite similar to those of Lyndon words in A*. For instance, m is
Lyndon if and only if m is primitive (i.e. m cannot be written m ~ pq, where
p and q commute and p, q / 1), and if it is the smallest element in its
conjugacy class (conjugation is the equivalence relation ~ in M generated by
the relations pq ~ qp, p, q 6 M). A technical lemma, which is not obviously
equivalent to the latter, is that m is Lyndon if and only if m < qp for any
nontrivial factorization m = pq.
Define IA(m) to be the set of a in Я such that m e aM. A pyramid is an
element m such that |M(m)| = 1. A pyramid m is admissible if IA(m) consists
in the smallest letter appearing in m. Each Lyndon element is an admissible
pyramid. The set of admissible pyramids m such that IA(m) = {a}, for a fixed
a in A, is a free monoid Ma, and an admissible pyramid m e Ma is a Lyndon
element in M if and only if it is a Lyndon word in the free monoid Ma.
Each element in M has a unique factorization into a decreasing product
of Lyndon elements. If m is Lyndon, not in A, then it has a unique nontrivial
factorization m = pq, where q is chosen minimum for the total order < in
M. This factorization is called the standard factorization of m. By iteration of
the standard factorization, one associates with each Lyndon element m a
binary tree; this tree, when the nodes are interpreted as the Lie bracket in
the free partially commutative Lie algebra cf(A, 0), defines an element of
У(А, 0). The set of all these elements forms a basis of cf(A, 0).
All these results are due to Lalonde (1992). His theory is expressed in the
geometrical language of heaps of pieces (see Viennot 1986). Conjugation in
7.6 Appendix
171
M(A, 3) was studied first by Duboc (1986). A consequence of this is that,
when A is finite, the dimension of the space of homogeneous elements of
degree n in ^F(A, 3) is equal to the number of primitive conjugation classes
of degree n in M(A, 3). This number bn may be computed as follows. Denote
by a„ the number of elements of degree n in M(A, 3). Then the generating
function of the a„ is given by the following formula, due to Cartier and Foata
(1969):
X anXn = fl - X ’
л > 0	\ л > 1	/
where cn is the number of subsets В of A of cardinality n and such that
ab = ba mod 3 for any a, b in В (see also Lallement 1979, Section XI.3). By
a result quoted above, there is a bijection between the set of elements in M
of degree n, and the set of decreasing products of Lyndon elements, of total
degree n, so that
Then, to compute the bn, one takes the logarithmic derivative and applies
Mobius inversion, as in the proof of Corollary 4.14. The same computation,
using the method of Witt (1937), gives the dimension of the space of
homogeneous elements of £f(A, 3); see Duchamp and Krob (1992c).
7.6.2 Irreducible polynomials over a finite field
Let F be the field with q elements. Then the number a„ of irreducible monic
polynomials in F[x] of degree n is equal to (7.1.1), a formula which was
known to Gauss. Indeed, F[x] is a unique factorization domain and there
are qn monic polynomials of degree n, so that
So one proceeds as in the proof of Corollary 4.14. This shows that the number
a„ is equal to the dimension of the space of homogeneous Lie polynomials
of degree n: this was noted by Witt (1937).
A direct bijection between primitive necklaces of length n over F and the
set of irreducible polynomials of degree n in F[x] may be described as
follows: let К be the field with qn elements; it is a vector space of dimension
n over F. There exists in К an element 3 such that the set [3, 34,..., 3qn
is a linear basis of К over F: such a basis is called a normal basis, and always
exists (see Lidl and Niederreiter 1983, Theorem 2.35). With each word
w = a0 ... a„_! of length n on the alphabet F, associate the element of К
172
7 Circular words
given by /? = ao0 +	+ • • • + an_ *. It is easily shown that to con-
jugate words w, w' correspond conjugate elements ft in the field extension
K/F, and that w i—► is a bijection. Hence, to a primitive conjugation class
corresponds a conjugation class of cardinality n in K; to the latter cor-
responds a unique irreducible polynomial of degree n in F[x], This gives
the desired bijection. Another bijection using, instead of a normal basis, a
generator of the cyclic group K O, is given in Golomb (1967).
7.6.3 Determinant of a sum of matrices
Given a square matrix x over a commutative ring, define the function A,(x)
by
det(l -00=1 + X (-l)VA.(x),
n> 1
where t is a commuting indeterminate.
Note that A„(xy) = A„(yx) for any matrices x, y. Let x15..., xk be square
matrices of the same size; we consider also {xn ..., xk} as an alphabet, to
simplify notations. For each primitive necklace v = (xfl... xlr), the matrix
function A„(v) = A„(xfl ... xir) is well defined. If M is a multiset of primitive
necklaces, let A(M) be the matrix function ||v AM(v)(v), where M(v) is the
multiplicity of v in the multiset M. Let sgn(M) be the sign of M, that is the
product of the signs of the necklaces in M, where the sign of (xfl... xlk) is
( — I)*-1 Then the following formula holds:
Л,(х1 +••• + xJ) = £sgn(M)A(M),	(7.6.1)
M
where the sum is extended to all multisets of primitive necklaces M of
length n over the alphabet {x15..., xk} (see Amitsur 1980; Reutenauer and
Schiitzenberger 1987). For example, A3(x + y) is the sum of eight terms,
given in Fig. 7.7.
For the proof, one uses the identity in Z<<Xp ..., xk>)
1 - Xi-------xk = П U - й)>
heH
obtained in the proof of Theorem 7.4, where H is a Hall set. Then one applies
the homomorphism sending the letter x, onto the matrix txo takes the
determinant, and uses the fact that Hall words are in one-to-one correspon-
dence with primitive necklaces (Corollary 7.5).
The Cayley-Hamilton theorem may be deduced from (7.6.1). Indeed, let
x, у be и by и matrices. Then A„+1(x + y) = 0; take in this equation the
terms of degree n in x, 1 in y, using (7.6.1). This gives
X (-1)'Л„.Хх)Л,(х'у) = 0.
1=0
7.6 Appendix
173
Fig. 7.7
Since At is the trace, we have, for any matrix у
tr( ( E (- l)iA„_,(x)xi)y) = 0,
\\i = 0	/ /
which implies the desired identity, by nondegeneracy of the trace. By taking
in the equation A„+	..., xn+ J = 0 (х,- n by n matrices) the multilinear
part, one obtains the multilinear version of the Cayley-Hamilton theorem
(see Procesi 1976, Theorem 4.3(b)).
7.6.4 Factorizations of the free monoid
A family (Xt)ie/ of subsets of A* is called a factorization of the free monoid
A* if / is totally ordered and if each word w in A* has a unique factorization
w = Xi ... x„,	x,. 6 Xjt, j\ > >]„.
(7.6.2)
In this sense, the family ({h})heH is a factorization of A*, for any Hall set H
(Corollary 4.7). A theorem of Schiitzenberger (1965)—see also Lallement
1979, Theorem XI.5.7; Lothaire 1983, Theorem V.4.1—asserts that for a
family of subsets (Xi)ieJ, indexed by a totally ordered set I, any two of the
174	7 Circular words
following conditions imply the third:
(i)	each word has at least one factorization (7.6.2);
(ii)	each word has at most one factorization (7.6.2);
(iii)	each submonoid X* is freely generated by Xi; conjugation within Xf
coincides with conjugation in A* and each conjugation class in A* meets
exactly one submonoid Xf.
Theorem 7.4 is a particular case of this theorem. The link between
factorizations of the free monoid and the free Lie algebra has been intensively
studied by Viennot (1978).
7.6.5 Zeta functions of cyclic languages
The zeta function of a language L (i.e. a subset of Я*) is
C(£) = exp ( X
\n> i n
where an is the number of words of length и in £ (we assume that A is finite).
For £ = (ab)*, i.e. the set of powers of ab, £(L) = (1/1 — t2)1/2 and for
£ = {a}, C(£) = e'.
A cyclic language is a subset £ of A* which is conjugation-closed and
power-closed; that is, uv & L о vu 6 £, and w 6 £ <=> wn e L, for any words
u, v, w and any integer n > 1.
The zeta function of a cyclic language has integer coefficients; in fact, one
has the formula
where a„ is the number of primitive conjugacy classes contained in £.
The main result of Berstel and Reutenauer (1990) is that if L is a cyclic
language which is recognizable by a finite automaton, then lj(L) is a rational
power series. For example, £(£) = 1/1 — t2, for £ = (ab)* и (ba)*.
7.7 NOTES
Formulas (7.1.1) and (7.1.2) must have been known for a long time. The first
formula of Corollary 7.3 appears in the book by Lucas (1891, p. 501), who
attributes it to Colonel Moreau. For the proof of Theorem 7.2, we have
followed Garsia (1990), with the help of Pierre Leroux. The algorithm of
Section 7.2 is from Melancon (1991), who extended an algorithm of
Meier-Wunderli (1951) and Schiitzenberger (1958). Corollary 7.7 is actually
7.7 Notes
175
the original definition of Lyndon (1954, 1955a). The results of Sections 7.3
and 7.4 are due to Duval (1978, 1983, 1988); see also Berstel and Pocchiola
(1992). The bijection of Theorem 7.20 is from Gessel in an unpublished
manuscript dated 1981, rediscovered by several people and published by
Gessel and Reutenauer (1992). Related bijections are in Metropolis and Rota
(1984), and Dress and Siebeneicher (1988). The standard permutation of a
word is introduced in Schensted (1961).
8
The action of the symmetric group
In Section 8.1 we present the duality between the representations of the
symmetric group and the linear group. From this, we quickly deduce the
character of the free Lie algebra, already knowing the generating functions
of primitive necklaces. This character is induced by a faithful one-dimen-
sional representation of the subgroup generated by a circular permutation
(Section 8.2). In Section 8.3 we give the combinatorial interpretation of the
multiplicities of irreducible representations in the Lie representation. It is
also shown that almost all irreducible representations actually appear.
Section 8.4 introduces remarkable Lie polynomials, the Lie idempotents. In
the last section, we consider the representations arising from the canonical
decomposition of the free associative algebra.
Throughout Chapter 8, we assume that К is a field of characteristic 0.
8.1 ACTION OF THE SYMMETRIC GROUP AND OF THE
LINEAR GROUP
Denote by K(Ayn the subspace of K{A) spanned by the words of length n.
There is a right action of the symmetric group S„ on defined by
(wa). =	i=l,...,n,	(8.1.1)
for any word w of length n, where w, denotes the г th letter of w. Equivalently,
if w =	... an, we have
(«i ... a„)o = a„(1)... a„(n).	(8.1.2)
This is indeed a right action, i.e. (wa)a = w(aa), for any a, a in Sn (see
Section 3.3). We call it sometimes the place permutation action of the
symmetric group, because the letters of a word are permuted according to
their position in the word. As an example, we denote by 2 3 1 the 3-cycle
(123) and have (abc)231 = bca and (aah)231 = aba (and not baa). This right
action of Sn on the words of length n extends by linearity to a right action
of the group algebra KSn on K{A~)n.
The group End( V) of endomorphisms of the К-vector space V = ®aeA Ka
8.1 Action of the symmetric group and of the linear group	177
acts naturally on the left on K<A>; each endomorphism of V extends
uniquely to a K-algebra endomorphism of KfA); the algebra endomor-
phisms obtained in this way are exactly the homogeneous algebra endomor-
phisms of K<A>. The previous extension from V to K<A> preserves
composition, as the universal property of implies, so that it is a left
action of End(E) on K<4>. This action preserves each submodule X<A>„.
These two actions on	commute with each other. Indeed, an
endomorphism f of Иis given by a matrix (ka b)a beA such that each column
has only finitely many nonzero elements and that
/(b) = X k.,ba.
aeA
for any letter b. Then the word w = br ... bn of length n is mapped by the
extension of f onto
/(w) =f(bl)...f(bn)
E	^01,bi • • • kantb„ai • • •
ai....aneA
Hence, for a in Sn,
f(w)a = X kai.bi • • •	• • • a<rn-
ai,..., ane A
On the other hand, we have
/(wct) = f(bal ...ban)
=	i b i * • • b Д1 • • •	*
ai...a„eA
Since ст is a bijection of {1,. .., n} with itself, we deduce (by the change of
variables a, = cai)
f(wa) = У kc , b , ... kc b cal ... c„n
•s \	Cal, Off I	"on о 1	url
Cl...cneA
= E kci.bi - kCn,bnc(T1 ...can=f(w)a,
Cl.....c„eA
because К is commutative.
A particular case of endomorphism of V is induced by a permutation of
A. Hence, we obtain an action of the symmetric group SA on the left on
K<A>„, given by
a«i • • • an = a(Ui ... a„) = «(aj ... a(a„).	(8.1.3)
More generally, each element of the group algebra KSA acts on the left
on K(A). When A contains {1, 2,..., n}, we have therefore two actions of
KSn on K<A>„, one on the right, one on the left. We call sometimes the left
178	8 The action of the symmetric group
action (8.1.3) the variable permutation action of Sn because the letters of a
word are permuted according to their actual value.
Still assume that A contains {1,..., n}. Denote by En the linear span of
all words w„ = <r( 1)... a(n), for a e Sn. Then £„, viewed as К-vector space,
may be identified with the group algebra KS„: under this identification, the
left (respectively right) action of a on a permutation a, identified with an
element of £„, corresponds to the left (respectively right) multiplication of a
with a in S„. More formally
= waa, waa = waa,	(8.1.4)
because (w^a), = (vvff)al = oai = (wffa),-, since (wff)j = oj by definition, and
(awff)i = «((wjj) = aoi = (w^f.
We identify £„ and KSn, and in particular a permutation with the
corresponding word; e.g. (134) (25) in S6 is identified with 354126. This allows
us to speak of Lie elements in KSn; e.g. 1 — (12) — (132) + (13) is a Lie
element KS3, because it is 123 — 213 — 312 + 321 = [[1, 2], 3] (we write 1
for the identity permutation).
It is important to note that the left action of Sn on £„ is equivalent to the
left action of Sn on its group algebra, hence the corresponding representation
of S„ is the regular representation.
We shall need in the sequel a result of representation theory, which relates
representations of the symmetric and the linear groups. Let £ be a subspace
of К {A) which is invariant under the previous left action, i.e. under each
algebra endomorphism of K(/l) sending each letter onto a linear combina-
tion of letters. Let n>0 and suppose that A contains {l,...,n}. We
call £ n En the multilinear part (of degree n) of £. The spaces £„ and £
are both invariant under the left action of Sn: hence we obtain a repre-
sentation of Sn on £„ = £„ n £. Denote by the character of this action.
The Frobenius image of or the characteristic ch(xn) of is the symmetric
function
ch(Xn) = -*. E Хп(ст)Рл(<7),	(8.1.5)
n! aes„
where л(ст) = z = Г‘2"2... is the cycle type of the permutation о (i.e. a has
n, cycles of length i) and рл = p" lfi2 • • • is the corresponding product of the
power sum symmetric functions Pi = x) (see Macdonald 1979).
Let a = (ай)оел be some multi-index, with the afl almost all zero, and
denote by Ea the space of finely homogeneous polynomials of partial degree
afl in each letter a. Let na = dim(£a n F) and consider the series
Z и, П u-eZ[[4]],	(8.1.6)
|a|=n aeA
179
8.1 Action of the symmetric group and of the linear group
in the commuting variables a 6 A, with
|a| = Iv
a
This series is symmetric in the variables a e A, because F is invariant under
any homomorphism sending each letter onto another one. We call it the
generating function of F n Hence we have associated with F and the
given integer n two symmetric functions of degree n (we look at symmetric
functions regardless of the set of underlying variables; this is possible if there
are at least as many variables as the degree of the symmetric function, here
n; see Macdonald (1979)).
Theorem 8.1 The symmetric functions (8.1.5) and (8.1.6) are equal.
This is the Schur-Weyl duality between the representations of the symmetric
group and the linear group (Weyl 1946, Theorem 7.6.F; Macdonald 1979, A7
in Chapter 1). The following example is typical of the use of Theorem 8.1.
Example 8.2 Let F be the free Lie algebra and n = 3. Then the multilinear
part £3 n F of F admits as basis the two Lie polynomials [[1, 2], 3] and
[[1, 3], 2] (see Section 5.6.2). Now, we have
(12)[[1, 2], 3] = [[2, 1], 3] = -[[1,2], 3],
(12)[[1, 3], 2] = [[2, 3], 1]
= [[2, 1], 3] + [2, [3, 1]] (Jacobi identity)
= — [[1, 2], 3] + [[1, 3], 2].
Hence the character y3 of the representation of S3 on £3 n F satisfies
Z3((l,2))=-1 + 1=0.
Moreover,
(123)[[1, 2], 3] = [[2, 3], 1]
= —[[1, 2], 3] + [[1, 3], 2],
(123)[[1, 3], 2] = [[2, 1], 3] = -[[1, 2], 3].
Thus,
Хз(123))= -1.
Evidently y3(id) = 2. Since there are 3 transpositions in S3, and 2 circular
permutations, we obtain by (8.1.5)
сй(Хз) = i(l -2-P? + 3 0-PjPj + 2(— l)-p3)
= 34p?-p3).	(8.L7)
180	8 The action of the symmetric group
On the other hand, F admits as basis a set of finely homogeneous
polynomials which is in bijection with Lyndon words with the same multi-
degree (Theorem 5.1). A word of length 3 in A*, with A totally ordered, is
Lyndon if and only if it is of the form
abc, a<b,a<c,b/c,
or
aab, a < b,
or
abb, a < b.
Thus, the generating function (8.1.6) is the sum in Z[[AJ] of the correspond-
ing monomials, that is
£ labc + £ (a2b + ab2).	(8.1.8)
a < Ь <c	a <b
To verify that (8.1.7) and (8.1.8) are equal, we compute (8.1.7):
(/	\ з \
(Eu) -Ед3)
\ a / a /
=	+ 3 E a2b + 3 E ab2 + 6 E abc-^a2]
\a	a<b	a<b	a<b<c	/
which is indeed equal to (8.1.8).
8.2 THE CHARACTER OF THE FREE LIE ALGEBRA
Let A be an alphabet containing {1,..., n}. Recall that the multilinear part
of degree n of the free Lie algebra &K(A) is the space &K(A) n En of Lie
polynomials that are linear combinations of words <t(1) ... o(n), a e Sn;
moreover, there is a left action of Sn on this space. We call Lie representation
of degree n or n-th Lie representation this representation of Sn. We denote by
Xn the corresponding character of Sn, and ch(xn) the characteristic of xn (see
Section 8.1).
Theorem 8.3 The characteristic of the representation of Sn on the multilinear
part of degree n of the free Lie algebra is
n dfn
(8.2.1)
Equivalently, for each permutation a in Sn, /„(a) = 0 unless a has only cycles
8.2 The character of the free Lie algebra	181
of length d, for some d dividing n, in which case
M ’ d^n/diWd).	(8.2.2)
n
We shall give three proofs of this theorem. The first one rests on the duality
between representations of the symmetric and the linear group (Theorem
8.1), while the other proofs are self-contained.
First proof of Theorem 8.3 (a) By Theorem 8.1, we have only to verify that
the generating function (8.1.6) of the space of homogeneous Lie polynomials
of degree n is equal to (8.2.1). But this space possesses as a basis the set of
Hall polynomials corresponding to Hall words of length n (Theorem 4.9(i)).
Observe that these polynomials are finely homogeneous, with same partial
degrees as the corresponding Hall words. Moreover, the Hall words of length
n are in evaluation-preserving bijection with primitive necklaces of length n
(Corollary 7.5) and the generating function of the latter is given by (8.2.1);
see Theorem 7.2.
(b) Denote by the number of permutations which commute with a given
permutation of cycle type z. If 2 = dn/d, then it is straightforward to show
that = dnld(n/d)\. Moreover, there are n\/zx permutations of cycle type z
in Sn, for each partition z of n. Hence, by (8.1.5) and the fact that /„ is
constant on conjugacy classes
ch(xn)= X
|A| =n
where Xn is the common value of %n of the conjugacy class consisting of
permutations of cycle type z. Thus
E XnZL'Px = 1 E P(d)pnd/d,
| A| = n	И d | n
by the first part of the proof. This proves (8.2.2) because the рл are linearly
independent. Conversely, (8.2.2) implies (8.2.1).	□
For the second proof of Theorem 8.3, we need two lemmas. The first one
is a result of linear representation theory.
Lemma 8.4 Let G be a finite group, e an idempotent in the group algebra
KG, and x the character of the action of G on the left ideal KGe.
(i) Then	E x(g~1)g= E xex"1.
geG	xeG
In particular, dim(KGe) is the product by |G | of the coefficient of 1 in e.
182	8 The action of the symmetric group
(ii) If G = Sn and e = E<res„ then the characteristic of % is given by
ch(x) = E a»Pw
rreS„
In particular, dim(KGe) =
Proof (i) Let g 6 G. Then %(g) is the trace of the linear endomorphism
u: x i—* gx, KGe -► KGe.
Denote by v the linear endomorphism
v: x i—* gxe, KG -► KG.
We have v\KGe = u, because e is idempotent. Moreover, if И is a subspace
of KG supplementary to KGe, e.g. KG(1 — e), then u(L) £ KGe. This shows
that
tr(v) = tr(u).
We compute tr(v). For this, let us compute first the trace of the
endomorphism wh: x i—> gxh, where h is in G. We have, because G is a basis
of KG: tr(wh) = number of x in G such that gxh = x; the latter equality is
equivalent to g~1 = xhx}. Hence, if we denote by (P, Q) the scalar product
in KG with G as orthonormal basis, we have
tr(wh) = E (xhx-^g-1).
xeG
Since
r = E (g /l)vvh,
he G
we deduce by linearity
X(g) = tr(v) = £ £ (e, h)(xhx'\g~v) = E (*?*" \ g~l),
heG xeG	xeG
which implies
E x(g)g~l = E
geG	geG
as was to be shown.
(ii) In Sn, g and g~1 are conjugate, so that we have by (i)
x(g)= E (xex~^g)= E (^x^gx).
xeG	xeG
183
8.2 The character of the free Lie algebra
Thus, by the definition of the characteristic of %,
ch(x)= 1 £ X(g)p^g) = * E x 1^)/’л(х >gX]= E (e' h)Pw)’
П. geG	g.xeG	he G
as was to be shown.
For the second lemma, let c, = (1 2 ... i) 6 Sn, for i = 1,..., n. In particular,
Cj = 1. Recall that we identify each permutation with the corresponding
word on {1,2,..., n] A, and KSn with the space En c К (A).
Lemma 8.5 For p = 0,..., n — 1, the following element of ZfA>:
с: -(-I)” E
n>ip>    ><2> ii > 1
is a linear combination of shuffle products иш v, u, v e A + .
The following example contains essentially the proof of this lemma.
Example 8.6 n = 5, p = 2.
c5 — С4С3 — e4C2 ~	— ^‘зС2 — C3C1 — C2CI
= 34512 - 34215 - 32415 - 23415 - 32145 - 23145 - 21345
= 34512 - (34 ш 21)5
= 34512 - 345 ш 21 + (345 ш 2)1
= 34512 - 345 ш 21 + 3451 ш 2 - 34512
= -345 ш 21 + 3451 ш 2.
We have used the dual identity of (1.4.2), defining the shuffle product.
Proof We write P = Q to express the fact that P — Q is a linear combina-
tion of и 111 v, и, и 6 A + . Then we have for any words u, v and letters a, b
(u ш vb)a = —(ua ш v)b,	(8.2.3)
by (1.4.2) and symmetry.
Let n > ii > i2 > • • > ip> 1. Then a straightforward induction, which is
left to the reader, shows that the permutation chci2.. .cip is of the form
upp ... u2 2 Uj 1 vn, where the word Uj has length ij — ij+ j — 1, with ip+l = 0
and where ... Ujt' = (p + l)(p + 2)... (n - 1). Hence, these permutations
are all distinct. Since they are ("~ *) in number and since they appear all in
the polynomial [(p + l)(p + 2)... (n - 1) ш (p ... 2 l)]n, which itself has
184
8 The action of the symmetric group
(" p1) terms, we deduce that
P= E. CiP---Ci2Ch
= [(P + l)(p + 2)... (n - 1) Ш (p... 2 l)]n.
Now, an iterative application of (8.2.3) shows that
F= -(\У(Р+ l)(p + 2)... (n — l)n 1 2 ... p = (-\yc£.	□
Second proof of Theorem 8.3 (a) Let Gn denote the intersection of En and
of the space generated by the elements и ш v, и, v e A +, and Fn the space of
Lie polynomials in £„. By Theorem 3.1, Gn and Fn are the orthogonal space
each of another, for the scalar product which admits A* as an orthonormal
basis. This scalar product is invariant under the left action of Sn on En, hence
the action of Sn on Fn is equivalent to the action of Sn on En/Gn. We compute
the character of the latter. For this, we may take К = C.
(b) Let £ be a primitive nth root of unity and define
A simple computation shows that e is an idempotent. The left action of Sn
on the left ideal CSne has a character /, whose characteristic is, by Lemma
8.4(ii)
ch(x) = - E ГкРл(С*) = 1 E E ГкРл(С*)-
П к = 0	ft d | n gcd(k, n) = n/d
Observe that the cycle type of c* depends only on gcd(k, ri) = n/d, and is
equal to dn/d. Moreover, gcd(k, ri) = n/d is equivalent to: £~* is a primitive
dth root of unity. Thus
ch(x) = - E Pdd E (1) = - E PdldP(d^ (8-2-4)
П d | n ы primitive	Udfn
d th root of 1
as is well known.
(c) Denote by = the equality mod Gn in En (identified with CS„). We have
by Lemma 8.5
=	(-D'C' L
ft p = 0	n> ip>    > i2> i i > 1
=	-c‘e2)(i -r'e,).
n
8.3 Irreducible components	185
Call и this latter element. Then и is invertible in CS„, because for p = 1,...,
n - 1, 0 / e - 1 = e - C”p = (C - cp)(Cp_1 + • • • + c^1), hence C - cp is
invertible, as is 1 — £ lcp.
Since Gn is invariant under the left action of Sn, we obtain
CS„e = CS„u = CS„ mod Gn.
Observe that CS„e and En/Gn both have dimension (n — 1)! (Lemma 8.4(ii),
orthogonality of Gn and Fn, and Section 5.6.2). Hence, the restriction to CS„e
of the canonical mapping En -> En/Gn is a linear isomorphism.
This shows that the left action of S„ on En/Gn is equivalent to that on CS„e,
and concludes the proof.	□
Corollary 8.7 Let a be an n-cycle in Sn and p: (ст) -> C a faithful representa-
tion of the subgroup generated by o. Then the representation induced by p to
Sn is equivalent to the Lie representation of degree n.
Proof Let co = р(ст). Then co is a primitive nth root of unity, and the
representation p is equivalent to the representation of (ст) on the (left) ideal
Kf of K(ct), with / = „ £*=0 co~kak, because
of = o)f.
Now, by definition of the induction, the representation of Sn obtained by
inducing p is equivalent to the representation of Sn on the left ideal KSnf.
With the notations of part 2 of the second proof of Theorem 8.3, we have
that e and f are conjugate idempotents. Hence, the characters of the
corresponding representations are the same, and their common characteristic
is given by (8.2.4). This concludes the proof, by Theorem 8.3.	□
8.3 IRREDUCIBLE COMPONENTS
Recall that for a standard tableau T of shape z(T) = 2, where z = (z15..., zk)
is a partition of n, a descent in T is an index i in {1,..., n — 1} such that
i + 1 is located in a lower row than i in T (in the English way of depicting
tableaux, i.e. rows increase in length from bottom to top). The descent set
of T is the set of descents of T, denoted by D( T), and the major index of T
is the number
maj(T) = £ i.
ieD(T)
For example, for the tableau T below, we have z(T) = (3,2, 1, 1),
186	8 The action of the symmetric group
D(T) = {2, 4, 6} and maj(T) = 12.
124
36
5
7
Recall also that the irreducible representations of the symmetric group Sn
are in one-to-one correspondence with the partitions of n, and that the
characteristic of the character corresponding to the partition A is the Schur
function (see Macdonald 1979).
The Lie representation has a special link with the representation of Sn on
a quotient ring of K[x15..., x„], which we study first. The action of Sn on
К [xb . .., x„] is given by
= P(xffl,..., xffn),
for any polynomial P in К [xb ..., x„] and any permutation a in Sn. Denote
by A(x15 ..., x„) the fixed subring of this action, i.e. the ring of symmetric
polynomials, and by I the ideal of R[xp..., x„] generated by the symmetric
polynomials without constant term. Let
R = KtX1,...,xnyi.
Since I is invariant under the action of Sn, R inherits the action of Sn.
Moreover, R inherits the graduation of K[x15..., x„]:
R = ф R,.
i>0
Theorem 8.8 The multiplicity of the irreducible character yf of S„ in its
representation on Rt is equal to the number of standard tableaux of shape p
and major index equal to i.
Proof (a) It is well known that K[x15..., x„] is a free A(xp ..., xn)-module
(see Bourbaki 1981b, Chapter IV, Section 6, Theorem 1). We show that there
is a K-linear isomorphism
R A(x)-+К [x].	(8.3.1)
Indeed, let (Pj)jeJ be a basis of R[x] over A. We show first that (Pj is a
basis over К of К [x] mod I. This is because each polynomial P may be
written P = PjQj for some symmetric polynomials Q,; hence, with a, =
constant term of Q}, we have P = a7P7 mod I. Now, suppose that ctjPj =
0 mod /; then ctjPj = QkRk for some symmetric polynomials Qk with-
out constant term, and some polynomials Rk, the latter may be written
8.3 Irreducible components	187
XjPjQkj for some symmetric polynomials Qkj, hence X;a;P; = XjPjYjcQkQkr
which implies = 0 because the Qk are without constant term and that (PJ
is a A(x)-basis.
Since R = K[x]//, we may define (8.3.1) by (P,-mod I) (x) Q i—» PjQ, and it
is indeed a К-linear isomorphism. This shows that for any choice of a basis
(Pj) of K[x]/7, the latter mapping is an isomorphism. Since I is invariant
under the action of Sn, and since this action is homogeneous, we may find
a homogeneous subspace of К [x] which is invariant under this action and
which is complementary to I. Take a homogeneous basis of this subspace:
then (8.3.1) preserves the grading and the action of Sn.
(b)	The isomorphism (8.3.1) preserves the grading and the action of Sn.
For a graded S„-module M = ®Mn, where each Mn is of finite dimension,
and for о in Sn, let us call generating series of the character of о on M the
series
n>0
where /„ is the character of Sn on Mn.
Then, the tensor product corresponds to the product of generating series.
We apply this observation to the isomorphism (8.3.1). The generating series
of ff on A(x) is П?=1 О — /)because A(x) is freely generated by the n
elementary symmetric functions e1,...,en, of degree l,...,n, as is well
known.
(c)	We compute the generating series of the character of о on К [xn ..., x„].
It is equal to £d>0 qd x (number of monomials xp = xf1 ... xPn left fixed by
о and which are of degree d). The action of о on xp is xp'(l)... xp”n
hence this monomial is fixed by о if and only if for any i and j in the same
cycle of ст,. one has p-t = pf, hence, fixed monomials of о are in one-to-one
correspondence with mappings f: {cycles of ст} -> N, and the degree of the
monomial is the sum
X /(c) x length (c).
ccycle of a
If ст is of cycle type z =	... лк, we deduce that the generating series of ст is
From (8.3.1), we thus have that the generating series of the character of ст
on R is
n	Ik
П (' -«')/ П (1 -<Л).	(8.3.2)
1 = 1	/1=1
188	8 The action of the symmetric group
In particular, for о = id, we obtain
Y dim R{ = (1 + <?)(1 + q + q2)... (1 + q + • • • + <-1),
i>0
which shows that К, = 0 for i > (2).
(d)	Let z be a partition of n. Then the symmetric function pA has the
following expansion in terms of Schur functions:
Pa = Z SpXA,
д
where Xa is the value of the irreducible character at a permutation of cycle
type z, and where the sum is over all partitions y. of n; see Macdonald (1979,
Chapter 1, (7.8)). Taking the value of these symmetric functions at 1, q,
q2,..., and using the identities
Jk 1
рл(1, q, q2,...) = П Рл,(1, 4,	• • •) = П ;----Г’
i = i	i = il-<T
and
s/l, q, q2,...) = Y qmai(T)l (1 - q‘),
т I 1 = 1
(where the sum is over all standard tableaux of shape /z; see Macdonald
(1979)), we find that (8.3.2) is equal to
Z Z 4mi'T'xi-	(8.3.3)
д А(Т) = д
Let nlfl denote the multiplicity of the character хц of Sn in its representation
on K,. Then (8.3.2) is equal to
i > 0	fi
By linear independence of the irreducible characters, the nifl are completely
determined by this equality. Hence, comparing (8.3.3) and the previous
expression, we get nifl = number of standard tableaux of shape у and major
index i.	□
Theorem 8.9 Let i be in the range 0 < i < n — 1, c an n-cycle in Sn, C the
subgroup generated by c and co a primitive n-th root of unity. The representation
of Sn on ®p = imOdn Rp is equivalent to the representation induced from the
1 -dimensional representation of C given by c —> co1. In particular, it depends
up to equivalence only on the subgroup generated by i mod n.
Using Corollary 8.7 and Theorem 8.8, we obtain the beautiful combina-
torial interpretation of the multiplicities of the Lie representation.
8.3 Irreducible components	189
Corollary 8.10 Let i, n be relatively prime integers. The multiplicity of the
character %* of Sn in the Lie representation of degree n is equal to the number
of standard tableaux of shape л and of major index congruent to i mod n.
We need to consider again, for each partition A = lai2a2... n"n of n, the
polynomial in q
nAq) = П
i = 1
1 -
(i-qT'
(8.3.4)
This is indeed a polynomial because
qn~^ + 1)
nAq) = itM
(1 -Oi -<T1)..-(i
1 - q*k
(8.3.5)
with A = (A*,..., AJ, A' = (zk_ 15..., AJ.
Lemma 8.11 Let в be a primitive p-th root of 1, where p divides n, and л a
partition of n. Then tiA&) = 0, unless /. = pnlp, in which case лАО) = Pnlp{u/p)\.
Proof Let z = (A15..., Ak). We have
(1-«)... (l-<f)
кАч) =----->---------
(1 -<z)...(!-q“)
The multiplicity of 6 as a root of the numerator is n/p; at the denominator,
it is equal to the number of A{ divisible by p; since £ A; = n, we conclude
that if лА@) t4 0, then each A; = p, thus A = pn/p. In this case, a simple
computation, using the identity
П (1 - «') = ₽.
i = 1
shows that ял(0) = pnp(n/pj\.
□
Proof of Theorem 8.9 Let /‘-denote the character of Sn induced by the
one-dimensional character a{ of C given by c i—> co1. Let a be a permutation
of cycle type A. By the proof of Theorem 8.8, the generating function of the
character of a on the spaces K, is given by (8.3.2), i.e. it is TiAq)- Thus, all
we have to show is that
л - 1
^Aq)= E mod qn - 1.
i = 0
(8.3.6)
It is enough to show that for each nth root в of unity, both sides of (8.3.6)
take the same vaue for q = в.
190	8 The action of the symmetric group
By Lemma 8.11,	= 0 unless в is a primitive pth root of unity, z = pn/p
and in this case ял(0) = p"/p(n/p)!.
We claim that if x is the characteristic function of the conjugacy class of
cycle type z in Sn, then Ео<г<л-1 ^‘<z‘, Z> vanishes, unless the following
conditions hold: 0 is of order p dividing n, к = p"/p; in this case this sum is
equal to 1.
Suppose this is proved. We have
<Z\ Z> = E z'(x)z(x) = ^z'ftf),
nl xesn
where nl/zA is the cardinality of the conjugacy class of cycle type z. Thus, we
find that 1 Eo < i < n -1 0*X*(v) vanishes, unless the previous conditions hold,
in which case the sum is equal to 1. Since z = pnlp implies zx = p"/p(n/p)!,
(8.3.6) is proved.
We prove now the claim. By Frobenius reciprocity
”E^<x\x> ='f^.xlQc
i = 0	i = 0
= E 011 E <^’)zW-
i = 0	n xec
This sum vanishes, unless z is the cycle type of some element in C, i.e. z = pnlp
for some p dividing n. Suppose that this holds. Since for x in C, x of cycle
type z <=> x of order p, and since c1 and co have the same order, we obtain
E0,<z\z> = -E E
i= 0	П i = 0 <i>J of order p
=1 E ”e
П a>J of order p i = 0
If £ is an nth root of unity different from 1, then Eo<i<n-i e‘ = 0, because
£ is a primitive <7 th root for some d dividing n and Eo<i<d-i e‘ = sum °f
all <7 th roots = 0. Hence, the previous sum vanishes, unless 0 is of order p,
in which case it is equal to 1. This proves the claim.
The last assertion of the theorem is a consequence of the following remark:
if i and j generate the same subgroup mod n, then coj = co\, form some
primitive nth root of unity со1Ф	□
The next result shows that, except in a few cases, every irreducible
representation of Sn occurs in the nth Lie representation.
Theorem 8.12 Let X be a partition of n. Then the irreducible representation of
Sn corresponding to A appears in the nth Lie representation if and only if A is
8.3 Irreducible components	191
not of the following form:
(i)	z = (n), n > 2;
(ii)	z = 1", n > 3;
(iii)	z = 22;
(iv)	z = 23.
By Corollary 8.10, this result is equivalent to the following corollary.
Corollary 8.13 Let i, n be relatively prime integers and a a partition of n.
Then there exists a standard tableau of shape a and major index congruent to
i mod n if and only if A is not of the form listed in Theorem 8.12.
The proof of Theorem 8.12 is rather technical.
Proof of Theorem 8.12 (a) We begin by the four cases. Let T be a standard
tableau of shape z. In case (i), maj(T) = 0. In case (ii), maj(T) = 1 + 2 + • • • +
n — 1 is congruent to n/2 if n is even, and to 0 if n is odd. In case (iii), T is
of one of the following tableaux:
1 2	13
34	24
whose major indices are 2 and 4 (the descents have been underlined). In case
(iv), T is one of the following tableaux:
1 2	1 2	13	13	11
34	3 5	2 4	2 5	2 5
56	46	56	46	36
whose major indices are 6, 10, 8, 9, 12. In all these cases, maj(T) is not
relatively prime to n, so that, by Corollary 8.10, the multiplicity of yf in the
nth Lie representation is equal to 0. hence the corresponding irreducible
representation of Sn does not appear.
(b)	From now on, we exclude cases (i)-(iv). We begin by the cases n = 1,
2, 4, 6, that is, we consider the partitions z = 1, I2, 31, 212, 51, 42, 412, 32.
321, 313, 2212, 214. For this, is is enough to exhibit a standard tableau of
shape z and major index congruent to 1 mod n (Corollary 8.10). These
tableaux are:
1	1	134	12	13456	1245
2	2	3	2	36
4
192
8 The action of the symmetric group
1 245	1 2 5	1 2 5	146	14	13
3	346	34	2	2 6	2
6		6	3	3	4
			5	5	5
					6
The reader may	verify	that in	each tableau the sum		of the descents
is = 1 mod n.					
(c)	It remains to show that in all other cases, the irreducible representation
VA of Sn corresponding to z appears in the nth Lie representation. By
Corollary 8.7 and Frobenius reciprocity, it is enough to show that the
restriction VA | C of VA to the subgroup C generated by an n-cycle c contains
a faithful one-dimensional representation of C.
Recall that for n / 4, the only nontrivial normal subgroup of Sn is the
alternating group. From this, we deduce that if an irreducible representation
of Sn is not faithful, then it is one-dimensional or n = 4. Thus, by 1 and 2,
we may therefore suppose that the representation VA is faithful on Sn.
(d)	Suppose that n = pr, p a prime. Then FJC is a sum of one-dimensional
representations of C; since the sum is faithful on C, one of these one-
dimensional representations is faithful on C (otherwise cpr~1 is mapped onto
the identity in the representation F., and the latter is not faithful). Thus,
we conclude in this case.
(e)	By (b), we may assume that n Ф 6. Hence, by (d), we may assume that
n = dq, d / 1, q = pr > 3, p a prime not dividing d. Since c is an n-cycle, cd
has d cycles, each of length q-, call them ol,...,ad. The group C acts
transitively on the set £ =	..., ad] by conjugation, and cd acts as the
identity of since q and d are relatively prime, cq generates C modulo the
subgroup generated by cd, and we conclude that cq cyclically permutes, by
conjugation, the elements of We thus may assume that
a. = c~q‘(T]Cq‘, i = 0,..., d - 1.	(8.3.7)
Let G be the commutative subgroup of Sn generated by Стр..., od. The
restriction VfG of V, to G splits into a direct sum of one-dimensional
representations of G. Since Ул is faithful on G and since cd is of order q = pr,
one of them is faithful on the subgroup of G generated by cd. Let x be the
character of this representation, and v a basis of this representation; hence
av = x(a)v f°r апУ in G. In particular, we have :v = /(ffjt', x(ai) = is
a gth root of unity, and x(cd) = x(&i ... trj = ^ ...	= £ is a primitive qth
root of unity.
(f)	For a in Sn, such that a~1Go £ G, denote by xf the one-dimensional
character of G given by /’(a) = z(<r Suppose first that £p ..., are
8.3 Irreducible components
193
not all equal. Then we can find a permutation i15.. ., id of 1,..., d such that
the cyclic permutations of the sequence , £id) are pairwise distinct.
Let tr in Sn be such that a~ 1oko = aik, for к = 1,..., d. Then
хЧъ) = /(<т~Ч<т) = /(ct.-J = £ik.
Thus, replacing / by /’, we may assume that the cyclic permutations of
(£i,  • •, ^d) are pairwise distinct. Since these d sequences coincide with the
d sequences (xcqk(ai))o<i<d-1 = (£;+jk)o<;><<! -1> by (8.3.7), with indices taken
mod d, we conclude that the d characters /'’’‘(О < к < d — 1) of G are
pairwise distinct. Hence, by a well-known theorem of Artin (see Lang 1965,
Theorem VIII.7), these characters are linearly independent.
Let и = v + e~ lcqv +  • • + e.~(d~ l)cq(d~ l’ve f<, where e is a primitive dth
root of unity. Then for a in G, we have
au= £ c~kocqkv
0 < к <d- 1
= £ s~kcqk(c~qkocqkv)
= £ E~kcqkx(c~qkacqk)v
= Y^kXc4k(^)c4kv.
Taking any linear form (p on V2, we obtain <p(ou) = ^с.~к(р(сцксУ/У‘,к(о).
Thus, if и = 0, we obtain <:k(p(cqkv) = 0, by linear independence of the
characters; hence cqkv = 0, which is not true.
We conclude that и / 0. Now, since cq is of order d, we have cqu = eu,
and since cdv = c,v, we have cdu = cu.
Since q and d are relatively prime, we deduce that си = сои for some (gd)-th
root of unity; but co must be primitive, because co4 = s (respectively cod = c)
is a primitive <7 th (respectively gth) root of unity. Hence, и is the basis of a
faithful one-dimensional representation of C contained in 1<, which proves
the theorem in this case.
(g)	Suppose now that = • • • = cd = 0. Then в is a primitive gth root of
unity, because so is ()d =
We may find numbers гг,... ,rd relatively prime to p, which are not all
equal mod q, and whose sum is d. Indeed, either p / 2 and we take = — 2,
r2 = 4, r3 =  •  = rd = 1 (then гг ф r2 mod q because q / 3), or p = 2 and we
take rx = — 1, r2 = 3, r3 = • • • = rd = 1 (then d > 3 and q > 4, hence r2 ф r3
mod q). Then of* is a g-cycle, and we may find a in Sn such that a~ 1oio = o'1,
1 < i < d; we thus have /’(o’,) = /(o’-	= /(of*) = 0r‘, and these d num-
bers are not all equal. Furthermore, xa(cd) = 0ri ... 0rd = 0d is a primitive gth
root of unity. Since /’ is a faithful one-dimensional character of the subgroup
generated by cd, we are back to (f) by replacing / by /’.	□
194
8 The action of the symmetric group
8.4 LIE IDEMPOTENTS
К is still assumed to be a field of characteristic 0. An element e of KSn is
called a Lie idempotent if e is idempotent and if the left ideal KSne is equal
to the multilinear part of the free Lie algebra over К on the alphabet
{1,..., n}, where as usual, each permutation is viewed as a word.
Theorem 8.14 Let e be a Lie idempotent in KSn, and Сл the conjugacy class
of cycle type /, in Sn. Then
v _Jn-1/z(p) if л = pnlp
aec, (0	otherwise,
where e„ is the coefficient of a in e.
Proof The theorem is an immediate consequence of Lemma 8.4(ii), of eqn
(8.2.1) in Theorem 8.3 and of the linear independence of the symmetric
functions рл.	□
Denote by 2\A)n the space of homogeneous Lie polynomials of
degree n.
Lemma 8.15 Let A be an alphabet with at least n elements, and e an element
in KSn. Then e is a Lie idempotent if and only if the linear mapping
n:K(A>n - K(Ajn,P^>Pe,
is a projection onto £P(A)n. In this case е = я(12...м), assuming that
{1,..., n} <= A.
Recall that Pe denotes the result of the right action of e on P (see Section
8.1).
Proof We may suppose that A contains {1,..., n}.
(a)	Let e be a Lie idempotent in KSn. Let w = ax ... an be any word of
length n in A, and define an algebra endomorphism (p of K(Aj> by (p(i) = a{,
(p(a) = 0 for any other letter. Then we have we = <p(12 ... n)e = <p(12 ... n e)
(by commutation of the left and right action on	see Section
8.1)	= (p(e), by (8.1.4). Thus we = <p(e) 6 LT(A)n, because (p(ff(A))	<f(A).
Any Lie polynomial of degree n is a linear combination of P = [cq ... an],
where [...] means some Lie bracket arrangement. We have Pe = [<p(l)...
<p(n)]e = <p([l ... n])e = <p([l ... n]e). Now, [1 ... n] is a multilinear Lie
polynomial, hence in KSne, by definition of a Lie idempotent. Thus
[1 ... n]e = [1 ... n], and Pe = <p([l ... nJ) = P. This shows that P i-> Pe is
a projection from onto ^(A)n.
8.4 Lie idempotents
195
(b)	Let л: P i-» Pe be a projection from К<Л>„ onto У(А)п. Denoting by
Fn the space of multilinear Lie polynomials, we have Fn = tt(Fn) = Fne £ KSne.
Moreover, KSne c KSn n n(KSn) c KSn n ^(A)n = Fn; hence, Fn = KSne.
In particular, e is a Lie element, and we deduce that e = л(е) = ее,
which shows by (8.1.4) that e is a Lie idempotent. Moreover, n(12 ... n) =
(12 ... n)e, by (8.1.4) again.	□
Recall that D(o) (respectively d(a)) denotes the set (respectively the
number) of descents of the permutation a; see Section 3.3. If S c
{1,..., n — 1} we denote by Ds the following element of KSn:
Ds= £ ff.
rreS„
D(a) = S
(8.4.1)
Theorem 8.16 The two elements of KSn below are Lie idempotents
.[[1,2]. 3],	-(I — (21))(1 -(321))...(I -(„...21))
n	n
= |ЛХ(-0‘В,1........(8.4.2)
n k = 0
(n- A-1	(-1)|S| (n- A-1
<reS„ n \ d(a) J S<=(1.7\n-1)	n	\ |S| J
(— 1)ISI
= X	+	<8A3>
Here we use the notation Ds for
DT,
T^S
(see Section 3.3). Note that in (8.4.2), we identify KSn with the space spanned
by multilinear words of length n on the alphabet {1, 2,..., n}.
Proof (a) The linear mapping itn, defined for any word ar... an of length
n, by 7tn(al ...an) = 1[.. .[[flj, a2f a3],..., a„], is by Theorem 1.4(v) (or
rather of its symmetric version), a projection from onto ^(A)n. By
definition of the right action, we have 7tn(al ... an) = (ax ... an)6n, with вп
defined in (8.4.2). Hence, by Lemma 8.15 л„(12 ... n) = if.. .[[1,2], 3].n]
is a Lie idempotent.
Observe that, by definition of the right action of Sn on the words of length
196	8 The action of the symmetric group
n, the right action of the cycle (n ... 2 1) on a word is given by
a j ... an(n ... 2 1) — апаг ... an _ x,
(8.4.4)
see (8.1.2). Thus, for any homogeneous polynomial P of degree n — 1 and
any letter a, we deduce by linearity
(Pa)(l - (n...2\)) = Pa-aP = [P, a].
This shows that, with the canonical embedding 8п_г -> S„, one has
я„(1 2 ... n) = IX-jO 2 ... n - 1), n]
= 7T„_1(12...n- 1)(1 — (n...2 1)),
and proves the second equality of (8.4.2) by induction, because kx(1) = 1.
Observe that if w is a permutation in Sn_^ with descent set {1,..., k]
{1,..., n — 2}, then the permutation wn in Sn has the same descent set, and
the permutation nw has descent set {1,..., к + 1}. Moreover, there are (" ~k ')
permutations in Sn with descent set {1,..., k], because such a permutation
is of the form f ... ik 1 Л ... j„_ x with h > • • • > 4, Ji < • ’ ’ < Л- i -*•
Since [.. .[[1, 2], 3],..., n] is of the form £ ±<t, with 2"~ 1 terms, all this
shows that this element is equal to	Z q( — 1 )kD{ x k} and it proves (8.4.2).
(b) Let Kj be the canonical projection -► LT(A), as defined in
Section 3.2. Let e = %x(l 2 ... ri). Then, by Lemma 3.14 and Corollary 3.16,
e is equal to any one of the expressions in (8.4.3).
It remains to show that for any P in К (/!)„, one has ях(Р) = Pe (then e
is a Lie idempotent by Lemma 8.15). But such a P is a linear combination
of polynomials of the form Q = <p(l 2 ... ri), where cp is an algebra endo-
morphism of К (A) which sends letters to linear combinations of letters. By
Lemma 3.9, cp commutes with я15 so that я1(2) = я1'>ф(12...п) =
(p - xfl 2 ... n) = cp(e) = <p(l 2 ... n • e) = <p(l 2 ... n)e (by the commutation
of actions—Section 8.1) = Qe.	□
We define the major index maj(a) of a permutation a to be the sum of its
descents:
maj(ff) =	'•
ie D(<t)
Theorem 8.17 Lew co be a primitive n-th root of unity in K. Then
кп = -
П creS,
is a Lie idempotent.
8.4 Lie idempotents	197
For example,
|(1 2 3 + j2 1 3 2 + j 2 1 3 + j22 3 1 + j 3 1 2 + 3 2 1)
is a Lie idempotent, for j a primitive cubic root of 1.
Lemma 8.18 Let S {1,..., n — 1} and P an homogeneous Lie polynomial
of degree n. Then
PD S = 0 if S / 0,
and in general
PDS = (- 1)IS|P.
Proof Let w = Uj ... an be a word of length n. Let S = {n15 пг + n2,...,
«!+••• + nk_ J, where the n-t are positive. Let 1 2 ... n = «j ... uk, where
each word u: is of length nh with nk = n — пг — • • • — nk _ P Then, by Lemma
3.13, we have
= 0(ui ш- • -ш uk),
where в is the linear endomorphism of KSn sending each permutation on its
inverse. Observe that by definition of the right action of Sn, we have
w(u1 Ш • • • Ш Uk) = Wj Ш • • • Ш wk,
where w = Wj ... wk, with |w;| = n-r
Thus, if S / 0, hence к > 2, we have by eqn (3.3.5):
(D^s, w) = (P0(Ui ш • • -Ш uk), w)
= (P, w(w1 Ш- • -LU икУ) = (P, Wj ш - • - ш wk) = 0,
by Theorem 3.1 (iv). This shows that PD S = 0-
Now, by inclusion-exclusion, we have
Ds= Z (-l)|s|-'^r.
This implies the lemma, in view of D0 =1.	□
Denote by c the cycle (1 2 ... n) in Sn.
Lemma 8.19 Let
1 "-1 . .
C„ = - £ W ‘c‘.
n i = 0
Then
Сл^Л ^лСл Cl’
and £п, кп are idempotents.
198
8 The action of the symmetric group
Proof Define d(a) by
J(ct) — d(a) + £,
where c = 0 or 1 according to a(n) < <t(1) or a(n) > In other words,
d(a) is the number of descents in a, viewed cyclically.
We claim that for a in Sn:
maj(tfc) = maj(tf) — d(a) mod n,
maj(ctf) = maj(tf) — 1 mod n,
d(ac) = d(a).
Suppose the claim is true. Then
Kn = 4 E Г wmaj('T) ’ ‘ac‘ = Д Г o)mai(ac‘ ’+ i(d(a) ~1 >ac!
n2aeSni = o	n2 a,i
= — E ^mai(ac,>^i(d(ac‘)~1)ffCi
n2 o,i
= A E wmaj(a,a J (a/,3,,“1);.
П meSn	i = 0
Now, for an nth root of unity p, one has E"=o Pl = 0, except when p = 1,
in which case it is n. Moreover, d(ct) = 1 if and only if i = c1 for some j, and
d(i) e {1,..., n}. Since maj(cJ) = n — j, for j e {1,..., n}, we obtain
1 n _. .
k'nSn ~ E
П j=l
Furthermore,
<Л = ‘/if E
И i = 0 a e S„
= \ E wrnaJ(c''T)^
n i.a
= 1 £ Wmaj,3,,7 = Kn.
tt meSn
Now, кп is idempotent, because кпкп = кп^пкп = с,пкп = кп, and similarly for
*эи'
It remains to prove the claim. Let о = аг ... an. Then ac = a2 ... anal,
which shows that
D(ac) = [i — 1 |z e D(a), i > 2} и E,
8.4 Lie idempotents	199
where E = 0 or {n — 1}, depending on whether an < or an > ax. Thus
maj(or) = J (z - 1) + (n - 1)|£| = J (z - 1) + (n - 1)|£|
ieD(a)	ieD(a)
i>2
= maj(tf) — d(o) — |£| mod n
= maj(o-) - d(o).
Also, co = (ai + 1)... (an + 1), where digits are taken mod n. If an = n,
then maj(ca) = maj(a) + n — 1 = maj(a) — 1 mod n. Otherwise, о = ... in
j..., co = ... (z + l)l(j + 1) • • • and the descent nj in о is replaced by the
descent (z + 1)1 in co; thus maj(ca) = maj(a) — 1.
The last equality of the claim is immediate.	□
For S {1,..., n — 1}, denote by maj(S) the sum of its elements.
Proof of Theorem 8.17 Let e be any Lie idempotent. We show that
екп = e.
This will show that the space KSne is contained in the space KSnKn. Now,
кп is idempotent by Lemma 8.19, hence the dimension of KSnKn is (n — 1)!,
by Lemma 8.4(ii). Similarly, the dimension of KSne is also (n — 1)!, hence
the two spaces are equal and кп is a Lie idempotent.
We have by definition of кп, by Lemma 8.18 and by the fact that e is a
Lie element:
eKn = e(1	£ comaj(S)Ds^ = - ( £ wmaj,s,(- l)|s|^)e
\n S c (1.Л- 1)	/ n \ S	/
= -( E wil + "'+Zk(-l)*je
П \0 < ii < • •  < ik < n	/
= - (1 — co)( 1 — co2)... (1 — co"- l)e = e,
n
because
(1 - co)(l - co2)... (1 - co"-1) = П (x - coz)|x=1
Z = 1
= (x"-1 + • • • + X + 1 )|x= j,
co being primitive.	□
Corollary 8.20 Let Hn be a set of representatives of the primitive conjugacy
classes of words of length n. Then the set {wk„ | w e Hn} is a basis of ^T(A)n.
For the definition of a conjugacy class, see Section 7.1. As the proof will
200	8 The action of the symmetric group
show, if a word w in Hn is replaced by a conjugate word w', then wKn is
replaced by w'k„ = Оюкп for some nth root of unity 0. This basis is therefore
canonical, up to constant factors which are roots of unity.
Proof Let be as in Lemma 8.19. Then c\n = a)^n. Hence, by this lemma,
скп = a)Kn. Two words w, w' of length n are conjugate if and only if w' = wc‘
for some i; in this case w'k„ = со1юкп. If w is not primitive, then w = c‘w for
some z, 1 < i < n — 1; then wk„ = o)‘wk„ => wk„ = 0.
These observations prove the corollary, because ^T(A)n = K^A^Kn is
spanned by the set {wk„| w word of length n}, and by equality of the number
of primitive conjugacy classes of length n and dimension of ^T(A)n (Corollary
4.14 and Theorem 7.1).	□
Third proof of Theorem 8.3 By Lemma 8.19, the mapping
К8„Цп ~~* KS„K„, X 1 * XKn
is a linear isomorphism. It is evidently an isomorphism of left S„-modules.
Hence, the two representations of Sn on these left ideals of KSn are
equivalent. The second one is the Lie representation, because кп is a Lie
idempotent. Hence, one concludes as in the second part of the second proof
of Theorem 8.3.	□
Let q be a variable, and denote by rn(q) the unique polynomial of
degree < <p(n) such that
tM = <T mod Ф„(з),
where Ф„(д) is the nth cyclotomic polynomial, and <p(ri) = degФn(^) is the
Euler function. Define
= - E rn(q)°-
U ae Sn
Theorem 8.21 Let f = <p(ri) — 1, and
Kn{q) = <50 + 6tq + • • • + dfqf.
Then
(i) Kn(q) is a Lie idempotent in K(q)Sn;
(ii) d0 +	+ • • • + cf()f is a Lie idempotent for any choice of scalars
ct,... ,cf in K.
Proof (i) Let e be a Lie idempotent in KSn. Then
KSne = KSnKn.
8.5 Representations on the canonical decomposition	201
This implies that екп = e and кпе = кп. Since, by Theorem 8.17, k„(cu) = кп,
we obtain
ек„(4) = e, Kn(q)e = Kn(q) mod Ф„(^).
Since both sides of each congruence are of degree < <p(n), we actually obtain
ек„(4) = e, Kn(q)e = Kn(q).
(8.4.5)
These equations imply that Kn(q) is idempotent and that K(q)SnKn(q) =
K(q)Sne, hence Kn(q) is a Lie idempotent.
(ii) Eqn (8.4.5) shows that ed0 = e, ed-t = 0 if i > 1, and 6-te = if
i > 0. Hence, e = e(<50 +	+ • • • + cf8f) and <50 + cl8l + • • • + cf8f =
(<50 + cl8l + • • • + cfdf)e. This proves (ii).	□
8.5 REPRESENTATIONS ON THE
CANONICAL DECOMPOSITION
Let z = Xj ... лк be a partition of n. We define Uk to be the subspace of
linearly generated by the polynomials
k'. aeSk
where each P; is an homogeneous Lie polynomial of degree z{.
Lemma 8.22 К<Л> =
In particular, with the notation of Section 3.2, let Uk be the subspace of
К<Л> generated by the kth powers of Lie polynomials. Then
Uk= ® Uk,
ia>=k
where 1(a) is the length of z (see Proposition 3.6).
Proof We may assume that the alphabet is finite.
Let (Ph)heH be a Hall basis of &(A). To each word w, which has the
decreasing factorization w = hx ... hk (see Corollary 4.7), associate the
polynomial
Qw = (Phi,...,Phk).
By multilinearity of the operator (P1?..., Pk), and by Theorem 3.7 and
Proposition 3.6, the polynomials Qw generate the vector space К<Л>. Since
they are in degree-preserving bijection with the words, they form a basis of
К<Л>.
202	8 The action of the symmetric group
The same argument shows that the Qw, for which (deg Phi,..., deg Phk) is
the unordered partition z, form a basis of If. Thus the lemma follows. □
Evidently, each If is invariant under each homogeneous algebra endo-
morphism of К<Л>; hence under the left action of Section 8.1, we obtain a
representation of Sn on the multilinear part of If, when |2| = n (see Section
8Л).
To describe this representation, we need the plethysm of symmetric
functions (see Macdonald 1979). Let x2,... be infinitely many commuting
variables, and g(xx, x2,...) a symmetric function in the xr Each symmetric
function has a unique expansion as a series in the power sums p2,.... If
g = G(p1? p2,...) and f = F(Pi, p2,   •) are two symmetric functions, then
the plethysm of f in g is defined to be the symmetric function
where f = F(Pi, p2i,...). When f has coefficients in N, there is an equivalent
definition. Let
(8.5.1)
16
where each m, is a monomial in the Xj. Then
g°f = д(тг,т2,...).	(8.5.2)
Since g is symmetric, the choice of the representation (8.5.1) is immaterial.
Observe that g i—► g ° f is for fixed f an homomorphism of the algebra of
symmetric functions into itself; furthermore, the mapping f pn ° f is linear,
for a fixed integer n.
We denote by hn the complete homogeneous symmetric function of degree:
hn is the sum of all monomials of degree n in the x-r i.e.
hn = Z	(8-5.3)
i 1 <   < i„
Theorem 8.23 Let /. = lni2"2... knk be a partition of n. Then the characteris-
tic of the representation of Sn on the multilinear part of If is
к
П V'-
I = 1
where ln denotes the characteristic of the n-th Lie representation, given by
(8.2.1).
Proof By the proof of Lemma 8.22, the space If admits as a basis the
set of polynomials (Plk,..., Plni, P2l,..., P2ni,..., Pkl,..., Pknk), where the
polynomials PtJ are chosen in some Hall basis of LF(A), with the condition
<teg(Pif) = i, and Pn < • • • < Plni, P2l < • • •< P2n2,..., Pkl < • • < Pknk,
8.5 Representations on the canonical decomposition	203
where the order is that of the Hall basis. The previous polynomial is finely
homogeneous, and Hall polynomials are in evaluation-preserving bijection
with Hall words. The generating function of the Hall words of degree n is
equal to Hence, by definition (8.5.2) of the plethysm and (8.5.3) of hn, the
theorem follows from Theorem 8.1.	□
Denote by ZA the centralizer of some permutation of Sn of cycle type 2.
The next theorem extends Corollary 8.7. We suppose that К contains all
roots of unity.
Theorem 8.24 Let 2 be a partition of n. The character of Sn on If is induced
by a one-dimensional character of Z;.
We call characteristic of an element e = ^aeSn eaff of the group algebra
KSn, the symmetric function ch(e) = Y^sn Ww-
Lemma 8.25 Let e, f be idempotents in KSn, KSP respectively. Denote, for о
in Sn, a in Sp, by о x a the permutation co in Sn+p such that: co(i) = a(i) if
1 < i < n, and co(i) = %(i — n) + n if n + 1 < i < n + p. Then
e x f =	x a)	(8.5.4)
<reS„
aeSp
is an idempotent in KSn + p, and
ch(e x f) = ch(e)ch(f).	(8.5.5)
Proof A straightforward computation shows that e x f is idempotent.
Observe that the cycle type of a x a is the union 2(a) u 2(a), hence
Pz(<7 X a) = Рл(<7)Рл(а)- This implies (8.5.5).	□
Lemma 8.26 Let e, f be idempotents in KSm, KSP respectively. For о in Sm,
ulf...,um in Sp, denote by (a; u1,..., um) the permutation of Smp defined (a;
и,,..., um) ((i - l)p + j) = (a(i) - l)p + Ui(j),for 1 < i < m, 1 <j < p. Then
e°f = Z ?afUi ...fUm((j;u1,...,um)	(8.5.6)
aeSm
ui...umeSp
is an idempotent in KSmp and
ch(e ° f) = ch(e)n ch(f).	(8.5.7)
In order to understand the previous definition of (a; u17..., um), consider the
partition
{l,...,mp} = IJ {(i-l)p +	—l)p + pj. (8.5.8)
1 < i < m
204	8 The action of the symmetric group
Then щ permutes the ith block of this partition, while о permutes the blocks
themselves.
Proof Let a, a e Sm, Up ..., um, vx,... ,vme Sp. Then
(a; u19, um)(a; , vm) = (aa; ..., ua(m)vm). (8.5.9)
Indeed, for 1 < i < m, 1 < j < p
(a; Up ..., um)(a; ..., vm)((i - l)p +j)
= (a; Up ..., um)((a(z) - l)p + vf(J))
= (aa(z) - l)p + u^v-Xj)
= (aa; ua(1)t?p ..., ux(m)vm)((i - l)p + j).
Since e, f are idempotents, we have
Z eeea = ev, £ fufv = fw.	(8.5.10)
ff(X = (p	uv — w
Now, we have
= Z e„fUi    fUmeJVi   	um)(cc; vx,..., vm)
where the sum is over all a, a in Sm, ub ..., um, vr,..., vm in Sp.
Hence, by (8.5.9), the coefficient of (<p; wb ..., wm) in (e f)2 is
Z	  • fvm Z ^a^x Z fu^iyfvi • • •
(JH = (p	(JO. ~<P	U2(i)Vi = Wi
Vi, = wt
= Z e^x Z frJV1 • • • frmfVm
m
= Z e^x П Z fr.fv,
ax = <p	i = 1 r,v, — wt
Z ^a^x J”[ fwi e<pfwi   • fwm’
ox — <p	i = 1
by (8.5.10). This proves that e°f is idempotent.
Let us view the set {l,...,mp} as {1,..., m} x {1,..., p}, and the
partition (8.5.8) as
x {l,...,p} = (J {z} x {l,...,p}.
1 < Z < m
We denote by Bt the z th block of this partition. With this identification, the
permutation (a; up ..., um) acts as
(a; Up ..., um)(i,j) = (a(z), u;(j))-
(8.5.11)
8.5 Representations on the canonical decomposition	205
We prove formula (8.5.7) without the assumption that e is idempotent; since
this formula is linear in e, we may assume that e is a single permutation:
e = e Sm.
Write a = Cj ... ck as a product of disjoint cycles; each cycle is of the form
c: = (rb..., r() and we denote u|c; = uri... uri, where и = (ub..., um) e S”.
We claim that
к
ch((o;ul,...,um)) = П Рк(°сИ(и\с{),	(8.5.12)
i = 1
where z; is the length of the cycle c;.
Suppose this is true; for и as above, let fu = fUi ... fUm and (a; u) =
(a; ub..., um). Then
ch(oof)= fu fl P^ach(u\c:) =	Z fu П Pk,°ch(Vi)
ueSp i ~ 1	vi, VkeSp ueS? i = 1
Vi,u|c, = v>
к
Z П p^° ch(vf Z fu-
vi,..., VkeSp i ~ 1	Vi,u|c, = t’,
Now, f is idempotent, so that fl = f for any I > 1, hence £W) ...W, = K,/W1 ...
fwt = fw- This implies that the last sum in the previous expression is equal
to fVi ...fVk. Thus
к	к
ch(a°f) =	ПАРл,°^.)=П Z ft’,Pi,°ch(Vi)
vi..Vke5pi = l	vteSp
к	/	\	к
= П Pit ° ch\ Z fv,vi) = П Рл, ° ch(f) = рЛ(а) ° ch(f)
i - 1	\vteSp	/	i - 1
= c/i(a)oC/i(/).
It remains to prove the claim. It is equivalent to show that 2((a; u)) is the
disjoint union of the partitions z,z(u|c;), i = 1,..., k, where the factor z;
means that each part of z(u|c;) is multiplied by z;. It is enough to show that
each cycle of length h of u|c; defines a cycle of length z;/i of (a; u); so let
С; = (r15..., r,), with I = /.h and c = (sb ..., sh) a cycle of u|c;. Then by
(8.5.11) the successive images of (rpsj under (a; u) are: (rb s,)(r2,
(r3, ur2uri(s i)),..., (rp un_, ... un(Sj)), (rp s2), (r2, uri(s2)),...,
uri(s2)), (rP sh), (r2, u„(sh)), ...,(r„uri_,... urj(sh)), (Гр sj.
Hence, we obtain a cycle of length fh of (a; u). These cycles are all distinct,
and their total length is £‘=1 z;p = mp, so that we have described all the
cycles of (a; u).	□
Proof of Theorem 8.24 (a) Suppose first that n = mp and tht z = pm. We
206	8 The action of the symmetric group
apply Lemma 8.26 to
1	1 p~1	. .
e = — Z a and f = ~ Z 0) >c‘’
ml aeSm	Pi-0
where co is a primitive pth root of unity and c the cycle (1, 2,..., p). Then
e ° f is in the group algebra KZ, where Z is the set of permutations in Smp
of the form (a; ch,..., ctm), with the notation of Lemma 8.26. In other words,
Z is the centralizer, in Smp, of the permutation (1, 2,..., p) (p + 1,..., 2p)...
((m — l)p + 1,..., mp), of cycle type A. We have
e°f = —?—	£	w"'1	‘m(a; c11,..., clm).
m\pm o<i....im<P
aeSm
The multiplication rule (8.5.9) shows that the ideal KZe° f is of dimension
1. Furthermore, the characteristic of the representation of Sn on KSne° f,
which is induced by the representation of Z on KZe ° f, is by Lemma 8.4(ii)
equal to ch(e° f). By Lemma 8.26
ch(e ° f) = ch(e) ° ch(f) = hm ° Ip,
because hm = Y^sm Piw as is wel1 known, and ch(f) = lp, by (8.2.1),
Corollary 8.7, and Lemma 8.4(ii).
(b) Let Z = Z; be the centralizer of some permutation о of cycle type к
Since conjugate permutations give conjugate centralizers, and equivalent
induced representations, we may assume without loss of generality that the
fixed points of a are {1,..., mr}, the 2-cycles of a act on + 1,..., тг +
2m 2}, and so on. Hence, we take a in the subgroup Smi x S2mi x S3m3 x • • •
of Sn, о = (cr 1? a2, a3,...), where op e Spmp is the product of the p-cycles of a
and is of cycle type pmp. It is well known that Z = Zj x Z2 x • • •, where Zp
is the centralizer of op in 8pmp. In part (a), we have constructed an idempotent
ep of KZp which is one-dimensional (i.e. dim(KZpep) = 1) such that the
characteristic of the representation of Spmp on KSpmpep is hmp°lp.
Using Lemma 8.25, we construct a one-dimensional idempotent e of KZ,
such that ch(e) = ПР ch(ep) = fL This implies the theorem, by
Lemma 8.4(ii), Theorem 8.23, and the fact that representations of Sn are
equivalent if and only if they have the same characteristic.	□
8.6 APPENDIX
8.6.1 Multiplicities
Denote by aA the multiplicity of the irreducible character /' of Sn in the
nth Lie character. Then aA is by Corollary 8.10 equal to the number of
standard tableaux of shape z and major index congruent to 1 mod n. For
8.6 Appendix	207
practical computations, one can use the formula
*a = - Z	(8.6.1)
П d | n
where %* is the value of the irreducible character /z at an element of cycle
type p. This formula is obtained by expressing the symmetric function ln in
terms of the Schur functions:
ln = E aASA>
|A| =n
and by using the identities
Рд = I X^A
A
and (8.2.1). Thrall (1942) and Brandt (1944) already give tables for these
multiplicities, up to n = 10. Davis (1958) gives explicit formulas for ал when
z has at at most two parts. In order to use formula (8.6.1), Foulkes (1959)
gives a rule to compute the numbers derived from the Jacobi-Trudi
identity:
^A ^®l(^Ai-i + j)l < i, j< n’
which expresses in terms of the complete symmetric functions hk.
For d a divisor of n, replace in the above determinant each hk by l/(k/d)!
if d divides k, and by 0 otherwise; then x'd„d is (n/d)l multiplied by this new
determinant.
When n is prime, then ал = ^(/t„ — Xa<">))- Since /(zn) is always equal to 1,
0, or — 1, we have that ал is the nearest integer to (n'1/^) (Brandt 1944),
i.e. the order of the irreducible representation corresponding to z, divided
by n.
Table 8.1 gives the multiplicities of the irreducible representation of Sn in
the nth Lie representation, for n = 1-7; they were computed by Thrall (1942)
and Brandt (1944). We also indicate the order of the irreducible representa-
tion corresponding to the partition z.
8.6.2 Lie invariants
Let the alphabet A have q elements, and let V — \aeA Ka. The group GL( V)
acts on К<Л> (see Section 8.1). A Lie invariant is a nonzero homogeneous
Lie polynomial which is left invariant, up to a scalar, under this action. For
example, with A = {a, b}, the Lie polynomials
[a, b] and [[a, [a, />]], [6, [a, />]]]
208	8 The action of the symmetric group
Table 8.1 Multiplicities of the irreducible represen-
tation of Sn in the nth Lie representation, for n = 1-7
(after Thrall (1942) and Brandt (1944)).
z	Л	Lie	A	A	Lie
1	1	1	32	5	1
2	1	0	321	16	3
I2	1	1	313	10	1
3	1	0	23	5	0
21	2	1	2212	9	2
I3	1	0	214	5	1
4	1	0	I6	1	0
31	3	1	7	1	0
22	2	0	61	6	1
212	3	1	52	14	2
I4	1	0	512	15	2
5	1	0	43	14	2
41	4	1	421	35	5
32	5	1	413	20	3
312	6	1	321	21	3
221	5	1	322	21	3
213	4	1	3212	35	5
I5	1	0	314	15	2
6	1	0	231	14	2
51	5	1	2213	14	2
42	9	1	215	6	1
412	10	2	I7	1	0
are Lie invariants (Magnus 1940). With A = {a, b, c}, the Lie polynomial
[[[[a, bf [a, c]], [[a, b], [b, c]]], c] + [[[[/>, c], [b, a]], [[/>, c], [c, a]]], a]
+	[c,b]], [[c, a][a,b]]], b]
is a Lie invariant, indicated by Wever (1949). He observed also that the
degree d of a Lie invariant must be a multiple of q, and that there are no
Lie invariants in the following cases: q = 2, d = 4; q = 3, d = 6; d = q > 3.
He gave a formula for the number of invariants (counted up to a scalar
factor) of a given degree, for q = 2. Burrow (1958) constructs a Lie invariant
for any q > 4 and any degree mq. m > 2.
The representation theory of the linear group shows that the number of
Lie invariants of degree d = mq is equal to the multiplicity, in the Lie
representation of degree d, of the irreducible representation of the symmetric
group Smq corresponding to the partition m4: by Corollary 8.10 this dimension
is therefore equal to the number of standard tableaux of shape mq and major
8.6 Appendix	209
index congruent to 1 mod d. In particular, Theorem 8.12 shows that there is
always a Lie invariant, except in the cases indicated by Wever.
8.6.3 Conjugation on circular permutations
Conjugation on the set of circular permutations defines a representation of
Sn. In this action, the stabilizer of a single circular permutation c is the
subgroup C generated by c. Hence, this representation is equivalent by the
one induced from the trivial representation on C. By Theorems 8.9 and 8.8,
we deduce that the multiplicity of the irreducible character in this
representation is equal to the number of standard tableaux of shape z and
major index congruent to 0 mod n.
8.6.4 Lie orthogonal idempotent
By Theorem 3.1 (iv), Q<A> is the direct sum Q<A> = ^(A) ф S, where S is
the subspace generated by 1 and the shuffle products и ш v, и, v nonempty
words. Let ti: Q<4> -» ^(A) be the projection corresponding to this direct
sum. Then n commutes with the homogeneous algebra endomorphisms of
K<A>, because &(A) and S are closed under these endomorphisms. For
each n > 1, we thus have by Lemma 8.15 that %(12 ... n) is a Lie idempotent
(this observation is due to Garsia). Some results on this idempotent are given
by Duchamp (1991); he shows in particular that the set of polynomials of
the form и ш nv, uv e и 1 (uv denotes here the concatenation of the
words и and r), is a basis of the multilinear part of degree n of S. More
general idempotents may be obtained by using the direct sum of Section 6.5.1.
8.6.5 On the idempotence of 0„
The fact that the element 0„ in (8.4.2) is idempotent (which is essentially
equivalent to Theorem 1.4(v)) is proved directly by Garsia (1990) as
follows: 0„ is a Lie element, hence by Lemma 8.18 and (8.1.4), we have
0nDs = ( — l)|s|0„, for any S c {1,..., n — 1}. Since
e. = "f <-')%......M,
Jk = O
we deduce
e.».= - <-!>*<-1)4 = e„.
nk = 0
A direct proof of the idempotence of 0n/n is given as follows by Wever (1949).
Let yn be the cycle (n ... 21). Then he shows, by the use of the Jacobi identity,
8 The action of the symmetric group
210
that
0п + еп.1Уп0п = 0,	(8.6.2)
where is as usual embedded in Sn. Moreover by (8.4.2)
on-1on = o2„-1^-yn)
= (n — 1)0„_1(1 — yn) by induction
= (n-l)0„.	(8.6.3)
Thus,
в2п=0п-1в-Уп)вп by (8.4.2)
= вп-1®п - 0п-1Уп®п
= (n - 1)0„ + 0n by (8.6.2) and (8.6.3)
= n0n.
Another direct proof is given by Specht (1948), who actually proves a
particular case of Lemma 8.18 (see his eqn (24), p. 375).
8.6.6 An idempotent for the canonical decomposition
Let n = mp and let A be the partition pm of n. Let Sm[Sp] denote the wreath
product of Sm by Sp: it is the subgroup of Smp normalizing its subgroup
Sp = Sp x • • • x Sp, naturally embedded in Smp. Equivalently, Sm[Sp] is the
set of permutations which leaves invariant the partition (8.5.8); hence, Sm [Sp]
is the set of permutations (a; u19..., um), a e Sm, ui9..., um e Sp, with the
notation of Lemma 8.26. Let
g = -—	У wrnaj('T)cr,
Pmm! aeSm[Sp]
where w is a primitive pth root of unity and maj(a) the major index of a.
Then g is an idempotent for If (i.e. If =	and g is idempotent).
Indeed, let
e = — X <*,/ = - S comaj(u)u.
ml (zeSm	P ueSp
Then maj(cr; ub ..., u„) = maj(ux) + • • • + maj(u„) mod p, because the per-
mutation (ст; ux,..., u„), viewed as a word, is the concatenation of words of
length p, each one of the form
к + u,(l), к + u;(2),..., к + u:(p),
for some i and k. This implies that g = e° f is idempotent and that
8.6 Appendix
211
ch(g) = ch(e) ° ch(f) (Lemma 8.26). Hence, the representation of Sn on the
multilinear part of Ux is equivalent to that on the left ideal KSng, by Lemma
8.4(ii), Theorem 8.3, Theorem 8.23, and the fact that ch(e) = hm, ch(f) = lp,
because f is the Lie idempotent кр of Theorem 8.17. Actually, K(A\g is
equal to Ux: indeed, let cp: denote the algebra endomorphism of K<A>
sending the letter j onto j + pi. Denote by f: the element Ф ,(./). Then it is
easily verified that
which implies that g is in U,. hence K(A\g = U, . by equality of dimensions.
8.6.7 Generalized Jacobi identities
The left to right bracketing of words defines a linear mapping /: KSn -> KSn,
defined by /(al ... an) = [.. .[al, a 2],..., an], for any permutation a in Sn
(considered as a word). The kernel of I is clearly a left ideal of KSn. Define
elements Oj(j = 2,..., n) of KSn by
0j= 12 ... n + J/(12 ... (J — 1))G+ l)...n.
Then Ker I admits as a basis the (n — l)(n — 1)! elements aOj, 2 <j < n,
a e Sn, aj = 1, where the product means the product in KSn (Blessenohl and
Laue 1988).
Indeed, recall Baker’s identity l(Pl(Q)) = [l(P)> K2)J> f°r апУ polynomials
P, Q (see Section 1.6.6). It implies that 12... J + .//(12 .. .j — 1) is in the
kernel of /, hence Oj is also, because Ker I is a right ideal in the concatenation
algebra. Hence, the (n — l)(n — 1)! indicated elements all lie in Ker /. Observe
that aOj is of the form a + an element in KT, with T = {a e Sn, al = 1}.
Thus, each element of KSn is congruent to an element of К Г mod Ker /. But
the (n — 1)! elements 1(a), aeT, are linearly independent mod Ker / (see
Section 5.6.2), which concludes the proof.
Examples of elements in Ker I are
12 + 21, 123 + 231 + 312, 1234 + 2143 + 3412 + 4321.
The two first ones are the defining identities of Lie algebras (antisymmetry
and Jacobi identity), while the third was observed by Wever (1949 (9a)); he
also observed that the sum of all permutations in Sn(n > 2), and the sum of
all even permutations (n > 3), are both elements of Ker / (Wever 1949,
Chapter 1).
8.6.8 Subspaces
Let Г be a complete binary, rooted, planary, unlabelled tree. Each labelling
in A of the leaves of T defines a tree t in the free magma M(A) on A (see
212
8 The action of the symmetric group
Fig. 8.1 The left comb.
Section 0.2), hence a Lie polynomial in ^(A), by intepreting each internal
node of t as the Lie bracketing. The linear span of all these Lie polynomials
is a subspace of ^T(A), stable under the action of the linear group (see Section
8.1). Denote by VT the multilinear part of this subspace: the symmetric group
Sn (n = number of leaves of Г) acts on VT, as observed by Barcelo and
Sundaram (1992). They show that VT is the whole multilinear part of degree
n of T£{A) if and only if T is the ‘left comb’ (see Fig. 8.1) or a tree obtained
from it by reflections around the nodes: the if part follows from Jacobi’s
identity, or from Theorem 1.4(v); the only if part follows from Theorem 5.7
because if T is not of the indicated form, then VT J271 = [J?(A), 2T(A)],
and ^f(A)\^f1 contains polynomials of any degree n > 1, by the theorem.
Among other results, they also show that if S, T are trees and S[T] is the
tree obtained by replacing each leaf of S by T, then the representation VS[Ti
is the wreath product of the representations Ts and VT; in other words, it
corresponds, via the characteristic map, to the plethysm of the characteristic
of VT into that of Vs. Moreover, they call generator of VT the Lie polynomial
obtained by labelling T with 1, 2,..., n from left to right, and denote it by
wr; they show that if ws, wr are both idempotents up to a scalar factor (e.g.
if S is the left comb), then so is ws[r].
Sundaram (1992) continues the study of the subspaces VT. She shows that
if Г = (Tj, T2), where TJ is the left comb with n; leaves, then the representation
VT is equivalent to the external tensor product of the representations VTl
and VT1 if and only if none of no n2 divides the other. Let S be any tree and
define the tree S' be as in Fig. 8.2. Sundaram studies properties of restriction
and induction of the representation Vs„. In particular, if S = ТЦТ], where
T, U are left combs with respectively m, n leaves, and m > 2 then
Т/ |Smn+n ____ fiz I C	\ fSmn + и- 1
FSn iSmn+n- 1 — V i лтл+л-2>> I
For an extension of this study, see Calderbank et al. (1992).
8.6 Appendix
213
8.6.9 A problem of ^-enumeration
In the proof of Theorem 8.17 we established that KSne^KSnKn for any Lie
idempotent e. For the reverse inclusion, we have used a dimension argument.
As shown by Bergeron et al. (1988), this inclusion leads to a problem of
enumeration of permutations by major index. Indeed, by Theorem 1.4(iii),
it is enough to show that д(кп) = кп® 1 + 1 ® кп. By eqn (1.4.1), we have
д(Кп) = E (К'л’ u ш r)M ® v-
U. V
Hence, we have to show that for any nonempty words u, v with n = |u| + |r|,
one has
£	оГа](») = 0	(864)
But it is known that, for an indeterminate q,
maj(vv)
weu lu v
maj(u) + maj(v)
Y
я
= Q
where [] denotes the ^-multinomial coefficient: = п!ч/р1ч(п — p)!q, with
nlq = 1(1 +<?)... (1 + q + • • • + qn~J); see Garsia and Gessel (1979). Thus,
(8.6.4) easily follows.
8.6.10 Conjugation
The symmetric function ln is invariant under the automorphism co of the ring
of symmetric functions if and only if n is not the double of an odd number
(Foulkes 1959). Recall that ш(рк) = ( — V)k~lpk, and that co sends a Schur
function onto the Schur function of conjugate shape (see Macdonald 1979).
The first equality, together with eqn (8.2.1), easily implies the above assertion.
214
8 The action of the symmetric group
8.6.11 Representation on the canonical projections
The symmetric group Sn acts on the multilinear part of degree n of the space
Uk, generated by the /с th powers of Lie polynomials (see Section 3.2). Since
Uk= © U„
ia>=k
Theorem 8.24 implies that the order of this representation is the Stirling
number of the first kind s(n, k) (Reutenauer 1986b). Denote by fn k the
characteristic of this representation. Then one has the following generating
series for the fn k:
E fn.ktk = expf e E
n, к	\i> 1 d\n ГН	/
(Hanlon 1990). This may be established by using Theorem 8.23, which
implies that
E fnjf = f E M') ° f E - vW1/) ,
\	/ \d | n П	/
and by using the identity
EV = exp( £ yf'pj,
\i > 1 I /
and the definition of the plethysm in terms of the power sum symmetric
functions.
The representation of the symmetric group on the multilinear part of the
space F = ©klf, where the sum is extended to all partitions whose parts are
1 or 2, contains each irreducible representation exactly once; this is because
the generating function of F is by Theorem 8.23 equal to
n — n - .
a 1 - a a<b 1 - ab
which is by an identity of Littlewood equal to the sum of all Schur functions
(see Macdonald 1979, p. 45). The same property holds for the shuffle
subalgebra generated by the words of length 1 and 2.
Consider the space Gk = ©; If, where the sum is over all partitions whose
parts are divisible by k; denote by ynk the corresponding character of Sn.
Then a result of Scharf (1991) states that xn/[(a) is equal to the number of
solutions in Sn of the equation a* = a; this proves a conjecture of Kerber
stating that this central function is actually a character.
8.7 Notes
215
8.6.12 Plethysm and derived series
Let (^f") denote the derived series of the free Lie algebra (see Section 5.3).
Denote by dnk the characteristic of the action of the symmetric group Sk on
the quotient ^nl^n+i, and by dn the sum ^kdnk. Then, a consequence of
Theorem 5.7 is that d{ is the following sum of Schur functions:
= sl,n- 1 •
n > 2
Furthermore,
dn = d1n —d1o- -od1 (n times),
and
'= Y ',= Y *"
n > 1 n > 1
For these results, see Reutenauer (1990).
8.7 NOTES
The first study of the free Lie algebra in connection with representation
theory (of the linear group) seems to be a paper of Thrall (1942). Formula
(8.2.1) giving the character of the Lie action appears in Brandt (1944),
although it is already implicit in the Witt formula (1937). For the second
proof of Theorem 8.3 we have used an argument similar to that of Lehrer
and Solomon (1986) in the proof of their Theorem (3.9). Another proof of
Theorem 8.3 is given by Joyal (1986), as an application of his theory of
analytic functors and their indicator series. Corollary 8.7 is from Klyachko
(1974); see also Blessenohl and Laue (1989). Theorem 8.8 and 8.9 and
Corollary 8.10 are due to Kraskiewicz and Weyman (1987); for the proofs,
we have followed Garsia (1990), who himself uses ideas of Stanley and of
Stembridge (1989). Theorem 8.12 is due to Klyachko (1974).
Theorem 8.14 is due to Garsia (1990); in the case of the idempotent (8.4.2),
it was also obtained by Blessenohl and Laue (1990a, 1991). The idempotent
(8.4.2) was introduced by Dynkin (1947, 1949), Specht (1948), and Wever
(1949); the first equality in (8.4.2) is due to the latter two authors, while the
second equality is due to Blessenohl and Laue (1991), and to N. Bergeron
(see Garsia 1990, Lemma 1.1): the element (8.4.3) is from Solomon (1968b);
see also Reutenauer (1986b) and Helmstetter (1989). The element кп is the
idempotent of Klyachko (1974), to whom are due Theorem 8.17, Lemma
8.19, and Corollary 8.20. Lemma 8.18 is from Garsia (1990). Theorem 8.21
is from Bergeron et al. (1988), who also proved Theorems 8.23 and 8.24 (the
latter was conjectured by Stanley).
216	8 The action of the symmetric group
It is interesting to note that the representation of Sn which is conjugate to
the nth Lie representation appears at various places: the top homology of
the lattice of (set) partitions of {1,..., n} (Hanlon 1981; Stanley 1982); the
Stanley-Reisner ring of this lattice (Garsia and Stanton 1984); the Orlik-
Solomon algebra of this lattice (Lehrer and Solomon 1986). Barcelo (1990)
gives a direct bijective construction between these representations; see also
Barcelo and Bergeron (1990); for similar work on the group Bn, see Bergeron
(1991).
9
The Solomon descent algebra
The convolution subalgebra generated by the projections of the free associa-
tive algebra arising from its graduation is also an algebra under composition,
and is canonically isomorphic with a subalgebra of the symmetric group
algebra, called the descent algebra; the latter was introduced by Solomon
for each finite Coxeter group. The canonical projections of the free associa-
tive algebra, arising from its structure of enveloping algebra of the free Lie
algebra, correspond to the primitive idempotents of the descent algebra; this
is established in Section 9.2, where some insight into the structure of this
algebra is also given. In the next section, we study various homomorphisms
of this algebra: one is a mapping into the ring of symmetric functions, and
its kernel is the radical of the descent algebra; another maps each descent
algebra into another one, and is a derivation with respect to the convolution
product. Elements of the descent algebra are also characterized, in the
symmetric group algebra, by their action on Lie monomials. In the final
section, we introduce quasisymmetric functions, which are closely related to
the descent algebra, and we give an application to the enumeration of
permutations.
9.1 THE DESCENT ALGEBRA
Let A be an alphabet. Recall that Q<?4> is a graded Q-algebra, and denote
by <□</!>„ the space of homogeneous polynomials of degree n. We thus have
<□</!> = © Q<4>„.	(9.1.1)
n > 0
Corresponding to this direct sum, we have for each n a projection
д„:О<Л> -» <□</!>,
which is the identity on Q</4>„ and which sends Q<?4>m, т Ф n, to 0. Recall
that in Section 1.5 we have defined an associative product on Endo(Q</l»,
the convolution, denoted by ♦. Let Г denote the convolution subalgebra of
End(Q</4>) generated by the qn; that is, Г is the linear span of the elements
218	9 The Solomon descent algebra
♦ • • • ♦ Qiky h,. • •, > 0. Note that the identity is not in Г. We may write
Г = © Г„,
n > 0
where Г„ is the linear span of the qix ♦ • • • ♦ qik with + • • • + ik = n: indeed,
such an endomorphism acts on homogeneous polynomials of degree n, and
sends the others to 0 (see the definition of the convolution).
A pseudocomposition of n > 0 is a /с-tuple H = (/ij,..., hk) of natural
integers whose weight \H\ = hx + • • • + hk is equal to n; the length of the
pseudo-composition is 1(H) = k. A composition of n is a pseudocomposition
without zeros. To each pseudocomposition H is naturally associated the
composition C(H) obtained by removing the zeros in H.
Given a pseudo-composition H as above, we denote by qH the endo-
morphism of Q<A> defined by
Qh = Qhl*' ’ '*Qhk-	(9.1.2)
Observe that q0 is the endomorphism e of Section 1.5, which sends each
polynomial to its constant term. Hence, q0 is the neutral element for the
convolution, by Proposition 1.10. This implies that for each pseudo-
composition H, one has
Qh = Qc(H)-	(9.1.3)
For example, g(3 j 0 2 0) = q(3 1>2). Given a matrix M = (m:j) (with 1 < i <, k,
1 < j < p) of natural integers, its row sum is the pseudocomposition
(mu + m12 + • • • + mlp, m21 + m22 + • • •
+ m2p,..., mkl + mk2 + • • • + mkp).
Similarly, its column sum is the pseudocomposition
(mM + m2 !-)-••• + mtl,m12 + m22 + • • •
+ mk2,... ,mlp + m2p + • • • + mkp).
The pseudocomposition associated with M is
H(M) = (mir,ml2,... ,mlp,m21,m22,.. .,m2p,... ,mkl,mk2,... ,mkp).
In other words, H(M) is the pseudocomposition obtained by reading the
entries of M, row by row.
Example 9.1
1
1
M =
0 3\
2 0/
9.1 The descent algebra	219
The row sum of M is (4, 3), its column sum is (2, 2, 3), and its associated
pseudocomposition is (1,0, 3, 1, 2,0).
Theorem 9.2 Let H, L be two pseudocompositions. Then the composition of
the endomorphisms qH and qL is given by
Ян° Ql = Y Яшм),
where the sum is extended to all matrices M whose row sum and column sum
are respectively H and L.
Example 9.3 Let H = (2, 2), К = (3, 1). Then the possible matrices are o)
and (f O) Thus, 2)° <?<з, i) = <?<i.1,2,0) + 9<2,o,i,i) = <7(i. 1,2) + ^(2,1.1) by
the theorem and (9.1.3).
The proof of Theorem 9.2 requires several lemmas. Recall that the k-fold
coproduct bk has been defined in Section 1.4.
Lemma 9.4 One has the formula
dk°qi= E (4mi ®-• ®
mi + ’ • • + mu = I
Proof The lemma simply expresses the fact that is degree preserving,
which is a consequence of eqn (1.4.1).	□
Recall that conct:	-> K<4> denotes the к-fold concatenation pro-
duct Pj ®	® Pk i-> Pj ... Pk (see Section 1.5). If (il,...,ik) denotes a
sequence of positive integers, all < N, we denote
cone,...K<4>®" -а К<л>, P, ®   • ® P„ P,,... P,K.
Lemma 93
<Vconcp = [conclit + 1.(P-I)t + 1 ® • • -®conct.2t.рк]Ь^р.
Proof We have, because is a concatenation homomorphism,
.cone,!?, ® •• ® P,) = <>,(?,... P,) = «.(PJ... Й.(РГ)
= ( E (Pl, Un ш-••-®и1Л L..
\U1	Ulk	/
X( E (pp’upl ш’ • ’ш MPJWP1 ® • • -® “pk )
\Upl....Upk	/
9 The Solomon descent algebra
220
by Proposition 1.8. Hence,
«VconCpCPj ® • • - ® Pp)
= £ (Pl, «11 ш • • • Ш Ulk) . . . (Pp, Upl Ш • • • Ш upk)
Ulj
X (uu ...Wp1)®-- ®(wlt...Wpt).
On the other hand, we have
(<5fc)®P(P! ® • • • ® Pp) = <5t(Pi) ® • • • ® <5t(Pp)
= ( E (Л, ш-• • ш рц)рп ® • • -® vlk I ® • • •
El’ll.I’lk	/
L (Pr. ">,1 ш - • - ш vpk)vpl ® • • -® vpk I.
\Vpl, Vpk	/
Thus,
[conclit + 1...(p-in+1 ® • • -® conct>2....pt] ° (<5t)®p(Pi ® • • • ® Pp)
= E (Л, fl 1 Ш • • • Ш vlk)... (Pp, upl ш • • • ш vpk)(vl! ... rpl) ® • • •
®(vlk...Vpk),
which proves the lemma.	□
Lemma 9.6 We have for natural integers f,..., iN, j\,..., jN,
(qit ®    ® qij (qjt ®---®qjJ
= \qit ®---®qiv if 0’i, ...JN) = (h,    Jn),
(0	otherwise.
Proof This is evident.	□
Lemma 9.7 3fp -3P = 3kp.
Proof This follows from Proposition 1.8 and the associativity of the shuffle
product.	□
For a in SN, we also denote by a the mapping
Q<A>® f	Pam ® • • • ® Pa(N}.
Lemma 9.8 conc(<T1 aN) = concN a.
□
Proof This is clear.
9.1 The descent algebra	221
Lemma 9.9 For endomorphisms j\,..., fN of Q</1) and a in SN, one has
оc (Л ® • • • ® fN) = (Li ® • • • ® Ln) ° °-
Proof Indeed
f °(f, ® • • -®АХЛ ®   -® PM) = a(fAPi)®   ® fM)
and
(Ai ®-  ®/<,»)«»№ ®   ® P„) = (Z,i®-  ® /.„MP., ®-"® P,x)
Lemma 9.10 For a in SN, one has
OcdN = <>N-
Proof By Proposition 1.8, we have
<т<5л(Р) = crl £	(P, иг ш • • • ш uN)ul ® • • -® uN
\ui u \ e Л*
= £ (P, Uj ш • • • ш uN)ual ® • • • ® uaN
= E (Л Utfi Ш • • • Ш	® • • • ® uaN
(because the shuffle product is commutative)
= E (P, 1’1 Ш- • • ш VN)Vi ® • • -® vN = dN(P). □
vi..... r,x e A*
Proof of Theorem 9.2 Let H = (hr,..., hk), L = (/p..., lp). Then by (9.1.2)
and (1.5.7)
qH-qK = conct (qhi ® • • -® qhk)' dk concp (qh ® • • -® qlp) dp
= comv(ghl ® • • -® qhk)
(conclit + 1.(р-1)Л + 1 ® • • -® conct 2t pk)
^®^(qh®---®qip)6p
(by Lemma 9.5)
222
9 The Solomon descent algebra
= conct о
E (сопс1Д + 1
 4- n i p = hi
(p- 1)4 + 1 ® ’ ’ ’ ® COnCt,2t, ...,pk)
nk i +   + nk p — hk
4Qnil®Qn2l®---®Qnkt®Qni2®---®Qnkp)
°	E 4m,, ®4m2, ® • • -® 4mk, ®4m,2® ' • ’® Qmkp
_mi i + - - • + mki = li	_
TH 1 p “I-	’ 4“ MJc p Ip
° »?- ° 6P
(because multiplication preserves the degree, and by Lemma 9.4)
= conck ° E (conc1Jt + 1..............(p-ixt + 1 ® • • -® concti2t.pt)
mj i 4- • • • 4- mj p = hi
mk i +    4- mu p = hk
mi i 4- • • • 4-mki = h
m i p 4“   4" mkp Ip
°(4m,,® 4m2,® • • - ® 4mkpWkP
(by Lemmas 9.6 and 9.7)
= COnC1<t + 1..(p-1)(t+l...k, 2k....pk°l E 4m,, ® Qm2t ® • • -®Qmkp )°^kp
(by associativity of the concatenation product)
= concpk°<7 J £ qmii ®qm2x ®- • ® qmkp b<5tp
\mu	/
(by Lemma 9.8, with о = l[/c + 1] ... [(p — l)/c + 1] ... /c[2/c] ... [p/c])
= СОПСрк J E 4m,, ®4m,2®’ ’ ’® 4m,p®421 ®’ ’ ’®4ткр )0<7°<5tp
(by Lemma 9.9)
= concpt э(е qmtt ® 4m,2®- • -® 4mk₽y tkp
(by Lemma 9.10).	□
Corollary 9.11 The convolution subalgebra Г of End(Q></l» generated by
the qn (respectively the subspace Г„ generated by the qH with \H\ = n) is closed
under composition.
Note that Г„ has a unit element, namely qn, and that Г = ®„>0 Г„, as
algebra (under composition). Given a permutation a in S„, its descent
223
9.1 The descent algebra
composition is the composition of n
C(ct) =
such that when viewed as a word, о = иг ... uk where each word is
increasing and of length c,, and к is minimal. For example, C(21534) =
(1, 2, 2). In other words, the descent set of о is S(C(a)), where С h-► S(C)
is the canonical bijection between compositions of n and subsets of
{1, 2,..., n — 1} given by
(Ci, . . . , Ck) —► {c15 Cj + c2,..., C‘i + c2 +  • • + ck_ j}.
Recall that for a subset S of {1, 2,..., n — 1}, we denote by (respectively
Ds) the sum in <Q>S„ of all permutations whose descent set is contained in S
(respectively is equal to S). The linear span in <Q>S„ of the 2n l elements Ds
is a subspace, freely generated by them; it is equal to the linear span of the
elements Ds. because of the triangular relations
= £ DT.
T^S
Denote this subspace of QS„ by
If C is the composition corresponding to S, we write Ac = DsS. Recall
that we have defined a right action of <0>S„ on the space of homogeneous
polynomials of degree n (see Section 8.1).
Corollary 9.12 The subspace of <0>S„ is a subalgebra of QS„. If |Л| > n,
then the linear mapping
£,^r„	(9.1.4)
for any composition C of n, is an anti-isomorphism of algebras. If
{1, 2,..., n} A, then the reverse mapping is given bvf*-+ f(12 ... n), for any
f in Гп.
Proof By Lemma 3.13(H), we have for any homogeneous polynomial of
degree n,
P\c = qc(P),	(9.1.5)
where we use the right action of <Q>S„ on <□</!>„.
The linear mapping
QS„ -+ Endo(Q</l>n), x^(P^ Px),
in an anti-homomorphism of algebras, which is injective if HI > n. Since T„
is, by Corollary 9.11, a subalgebra of End(Q</l», we deduce that is a
subalgebra of <Q>S„ and that (9.1.4) is an anti-isomorphism if |Л| > n.
224	9 The Solomon descent algebra
Suppose that A contains {1,..., n}. Then, by (8.1.4), we have (12... n)
Дс = Дс, once more viewing each permutation as a word. This proves the
last assertion.	□
Denote by the subspace &(£„) of where в is the anti-auto-
morphism of <0>S„ sending each permutation on its inverse. By Lemma 3.13(i),
is linearly generated by the elements Uj ш-• ш for all possible
concatenation decompositions 12 ... n = щ .. .uk.
Corollary 9.13 The subspace	of Q>S„ is a subalgebra, isomorphic with T„.
Corollary 9.14 If A is infinite, then the convolution algebra Г is freely
generated by the qn, n> 1.
Proof By Corollary 9.12 the elements qc are linearly independent, because
so are the Дс.	□
9.2 IDEMPOTENTS
By Lemma 8.22, we have a direct sum decomposition
Q<4> = © If,	(9.2.1)
where the sum is over all partitions z, and where, for z = (z15..., Ak), If is
linearly spanned by the polynomials
(P], . . . , Pt) = ~ У Pa(l)  • • Pa(k)’
Kl aeSk
for each choice of homogeneous Lie polynomials of respective degree
z15..., zt. We denote by л/ <□</!> -*•<□</!> the projection onto If, cor-
responding to this direct sum.
Let T = {f1512,..., tn,...} be a new alphabet, and x, x15 x2,. .., x„,...
central variables. To each word w in T* is naturally associated a composition
C(w); we therefore denote by /(w) and |w| the length and the weight of C(w),
respectively. As an example, w = t2t2t3, C(w) = (2. 1, 1. 3), and l(w) = 4,
|w| = 7. We define elements Mn and Kk of Q<T> by the following generating
series:
У xnM„ = log( 1 + txx + t2x2 -I----1- t„xn -I-).	(9.2.2)
n > 1
У xkKk = expCxjMj -I- x2M2 +    + xnMn +   f. (9.2.3)
9.2 Idempotents	225
where for z = Г'2"2..., we denote xA = x'px"2 • • • • Observe that K{n} = Mn.
Example 9.15 Mx = M2 = t2 — jtf M3 = t3 — |гхг2 — 2t2t! 4- 3tf and
K(3) = M3, K(21) = 2MrM2 + \М2МХ =	+ 2^2^ 1 — 2G5 ^(1’) = 6-^1 =
1,3
6‘ 1 •
We define an algebra homomorphism from Q< T> with concatenation into
Г with convolution:
С-О<Г>^Г, Ш =
Thus, we have С(гп ... tik) = q(l *  -*qik, or equivalently C(w) = qC(w), for
any word w in T*. Note that C is surjective, and is an isomorphism if A is
infinite, by Corollary 9.14.
Theorem 9.16 The projections лА belong to Г, and are given by
=	(9.2.4)
The projections are mutually orthogonal idempotents of the composition
algebra Г, and their sum for |z| = n is the identity ofTn.
We need the following result, where 2T(A)l denotes the space of homogeneous
Lie polynomials of degree I.
Lemma 9.17 Let л = (zh ..., zt) be a partition and let r15... ,rk be endo-
morphisms of К (A) such that 1т(г,) 2T(A)Xi. Then the image of r — '£taesk
ral *• • -*rak is contained in If.
Proof We have by (1.5.7) and by Proposition 1.8 for any polynomial P
r(P)= £ conck (rffl ® • • -® rak) bk(P)
<reSk
= £ COnCj'(/‘(T1 ® • • -® r^)!	£ (P,Ul Ш- • -Ш ujuj ® • • -® uk
aeSk	\ui....ui<
= £	£ (P, Uk ш---шик)га1(и1)...гак(ик).
aeSk u 1.Ur
Since the shuffle product is commutative, the products rh(ujt)... rik(ujk)
corresponding to the same multiset [uh,...,ujk} all appear with the
same coefficient in this sum. Hence, this sum is a linear combination of
(r^u J,..., rk(uk)), and therefore in L/z, by hypothesis.	□
Proof of Theorem 9.16 With the notation of Section 3.2, we have Uk =
and therefore nk = £/(А)=л ял. In particular, щ = £„> j я(п).
226
9 The Solomon descent algebra
Furthermore, л(п) = q„ ° because £ &(A)„. Thus, by (3.2.3), for
any word w
(-I/-1	„
Ww) = E —7—	E <w’	w*)wi • • • uk
k> 1 К U1..........uke4 +
|u 1 . • -Ufcl =n
у (-1Г1
*>1 к
У	£	(w, Uj ш- • -шик)
. , Uk^A* 7, • • ., ik > 1
11 + • • - + ik = n
x сопс4 ° (qh ® • • • ® qik)(ul ® • • • ® uk)
= E — E сопск°(дг1 ® • --® qik)°bk(w)
к>1 К h............,\>l
i i +   + ik = n
(—I?-1
= E 1—i—	E (^h*-'-*^kXw)’
к > 1 К it.......ik > 1
by Proposition 1.8 and (1.5.7). On the other hand, by (9.2.2)
hence,
(__ I)*-1
C(M„)=	£	—--------<?,,*• • -*qik.
k, ii...... ik > 1 К
ii +   + ik = n
Thus, 7t(n} = C(M„). In particular, Im(n(n}) £ <?(А)Й.
By (8.2.3), we have that for z = (z15..., zt), is, except for a constant
factor, equal to the sum of the product MX) ... MAk and its permutations.
Applying C, we find that C(KJ is of the form indicated in Lemma 9.17, and
we conclude that its image is in Ux.
Now, by putting x = 1 in (9.2.2) and xt — 1 in (9.2.3), we find that
Ел = 1 + ^ + t2 + ‘ ‘ ‘ • Applying C we deduce that C(K;) is the
identity of K<A>. Thus, by (9.2.1), we deduce that C(KA) =
The remaining assertions are immediate.	□
Since the are idempotents which decompose the identity of the algebra
under composition Г„(|2| = n), we have the direct sum
Г„ =	©	(9.2.5)
Ц| =|д| -n
We shall compute the dimensions of these subspaces of Г„. For this, we
decompose the space Q<T> into a direct sum, similarly to (9.2.1), but using
weight instead of degree.
9.2 Idempotents
227
To each composition C (respectively to each word w in Г*), is naturally
associated a partition, denoted by 2(C) (respectively 2(w)). If 2(C) = 2, we
say that C is compatible with 2. Since a nonzero finely homogeneous
polynomial P 6 Q<7> is a linear combination of words in T*, all associated
to the same partition, we may denote by 2(P) this common partition. On
the other hand, such a polynomial has a weight |P|, equal to the weight |z(P)|
of the partition 2(P).
For two partitions 2, p, define a subspace M of Q<T>: it is generated
by the polynomials
(9.2.6)
where each Pj is a nonzero finely homogeneous Lie polynomial in ^f(P), with
2(Pi ... Plw) = 2
and
/((IPJ,..., |P/(M)|)) = p.
As an example, the subspace Из211 43 contains the polynomials ([tb t3],
[г15г2]) and (t3, [ti,[ti, t2]]).
We say that a partition 2 = (z15..., 2t) is finer than a partition p =
(p^ ..., ph) if for some partition {1,..., к} = Ex u • • • u Eh, one has p{ =
YsjeEi^j. We denote this by 2 > p. For example, 3211 >43. From the
definition of the subspace д, it follows that
В<_д/0=>2> д.	(9.2.7)
Let H be any Hall set in T* (see Chapter 4). Recall that each word w in T*
has a unique decreasing factorization
w =	... hk,	H, hj > • • • > hk.	(9.2.8)
Besides the partition 2(w) associated with w, we associate with w another
partition p(w), its type, which is the partition associated with the compositon
(|/ii|,..., |hfc|), and which depends of course on H.
As an example, H is the set of Lyndon words on T, naturally ordered.
Then w = t3t2t4t3tit2 has the decreasing factorization w = (ti)(t2t4ti)(t1t2),
and 2(w) = (4, 3, 3, 2, 2, 1), p(w) = (9, 3, 3).
Let Qw denote the polynomial Qw = (Phi,..., Phk), where Ph is the Hall
polynomial corresponding to the Hall word h; see Section 4.2.
Lemma 9.18 The polynomials Qw, with 2(w) = 2 and p(w) = p,form a basis
of . In particular, one has the direct sum
Q<r>= © W^.
228	9 The Solomon descent algebra
Proof It was shown in the proof of Lemma 8.22 that the Qw form a basis
of Q<T>.
Since each Hall polynomial Ph is finely homogeneous and has the same
partial degrees as the Hall word h, we have z(Ph) = 2(h) and |Ph| = |h|;
hence, X(Phi ... Phk) =/fhi .. .hk) = a(w) = л, and z((|Phl|,..., |PhJ)) =
zOil,..., \hk I)) = g(w) = g. Thus, Qw e
Now, each finely homogeneous polynomial is a linear combination of Hall
polynomials having same partial degrees as it. Thus, by multilinearity and
symmetry of the operator (,...,), we conclude that д is generated by the
Qw with z(w) = z and g(w) = g.
The last assertion follows immediately.	□
In eqn (9.2.2), we have defined elements M„, n > 1, of Q<T>. They are given
by
Define a (concatenation) algebra endomorphism у of K(Ty by
7(t„) = M„,n> 1.	(9.2.9)
Then, since Mn = tn + a Q-linear combination of words involving only the
Гг with i < n, we conclude that у is an automorphism of Q<T>.
Define an algebra homomorphism if from Q<T> onto the convolution
algebra Г by if = C ° 7- Observe that, since у is weight preserving, f maps
Q<T>„ onto T„.
Theorem 9.19 For any two partitions f, g of n, we have
= СЖ.Д
Before proving the theorem, we derive several consequences.
Corollary 9.20 Let H be a Hall set in T*, let z, g be two partitions of n,
and suppose that |4| > n. Then the dimension of лдГплА is equal to the
number of words w in T* such that z(w) = z and g(w) = g. In particular,
dim(nAr„nA) = 1.
Proof We have seen in Section 9.1 that the space admits as a basis the
elements Ac, C composition of n. Since the linear mapping Ac ь-► qc is by
Corollary 9.12 an isomorphism from onto Tn, we conclude that the qc
are linearly independent for |C| = n.
Hence, the linear mapping C, when restricted to the space Q<T>„ of
homogeneous polynomials of weight n in Q<T>, is an isomorphism onto
9.2 Idempotents	229
Г„. Since у is a weight-preserving automorphism of and С = C 7, we
conclude that is an isomorphism Q<T>„ -»• Г„. Thus, the first assertion is
a consequence of Theorem 9.19 and Lemma 9.18.
Suppose now that for some word w in T*, we have z(w) = p(w) = A. Then
with the notation of (9.2.8), ht must be a word of length 1 of each i; hence,
if z = (Л15..., zfc), z15..., zfc > 1, we have by (9.2.8) that w is equal to the
unique product of the rAt which is decreasing in the order of H. This shows
that dim(B< A) = 1, which implies the last assertion by Theorem 9.19.	□
Given a multiset of primitive necklaces M on the alphabet Г, we define z(M)
and p(M), similarly to what we have done previously: each letter in M
defines a part of A(Mf each necklace in M, which is the conjugation class
of L,... tik, defines the part ц + • • • + ik of p(M).
In view of Theorem 7.17, Corollary 9.20 implies the following result.
Corollary 9.21 Let |Л| > n.) Let A, p be two partitions of n. Then the
dimension of лдГп is equal to the number of multisets M of primitive necklaces
on T such that A(M) = z and p(M) = p.
Denote by p(n) the number of partitions of n and recall that the number
of compositions of n is 2"-1. By (9.2.5), (9.2.7), and Theorem 9.19 we
immediately have the following result.
Corollary 9.22
(ii)
(i) If A < p then лдГплА = 0.
Tn ©	71дГп71^.
x > д
|Л| = |д|=п
Corollary 9.23 The radical of T„ is equal to ©л>д ядГ„ял. Z/MI n> then
T„ is of dimension 2"-1 and its radical of dimension 2"-1 — p(ri).
Proof Let R = ®л>д ЛдСЛл and S = ©лГл, where the sum is over all
partitions of n. Since the are orthogonal idempotents, Corollary 9.22
implies that R is an ideal in T„ such that some power of R vanishes. Hence,
R is contained in the radical of T„. Now S is a semisimple algebra, being
isomorphic to some power of the field Q. By Corollary 9.22, the canonical
projection Г„ -► S (according to the direct sum T„ = R © S, by (9.2.5)), is a
homomorphism of algebra. Since its kernel is R, we deduce that Tn/R is
semisimple, hence R contains the radical. Finally, R is the radical of T„.
If |Л|^и, then T„ is of dimension 2"-1: indeed, so is having a basis
indexed by subsets of {l,...,n — 1}; moreover, they are isomorphic by
Corollary 9.12. Now 5 is of dimension p(n), hence the corollary follows. □
For the proof of Theorem 9.19, we need several lemmas.
230	9 The Solomon descent algebra
Lemma 9.24 Let QY,... ,Qp be Lie polynomials and к > 1. Then
•Wi. •  •, e,)) = L <Q,.) ® • •  ® (Q,
where the summation is over all к-tuples (/b..., Ik) such that {1,..., p] =
f u  • • u Ik (disjoint union), with
(в-) = ~ E ne.w.
\1 |! aeS, iel
Note that the last product is in the natural order of I.
Proof We may consider Qi,...,Qp as noncommuting variables. The
equality in the lemma is symmetric in QY,..., Qp, so it is enough to check
taht the coefficients of Er ®	® Ek on both sides are equal, for each choice
of a factorization . Qp = Ex . .. Ek.
Now, at the left this coefficient is, by Proposition 1.8, equal to (p!)-1
multiplied by the number of permutations о e Sp which appear in the shuffle
product Er ш • • -ш Ek; that is, by definition of the shuffle product
£____P'_______
p!|Ei|!.7.|£fc|!’
On the right, the tensor Er ®	® Ek appears only once, namely in the term
with Ij = set of indices of the Q{ appearing in Ej (and then |/y| = |£;|), with
coefficient
1 1
|/il’''' 141?'
This proves the lemma.	□
Recall that to each composition C is canonically associated a partition x(C).
For two compositions С, С', /.(C) = z(C') means that C' is a rearrangement
of C.
Lemma 9.25 Let Qi,... ,QP be homogeneous Lie polynomials in <Q><4> of
degree dlf..., dp. Then for f,.. . ,ik> 1, the polynomial (n(il) *  • • * 7t(ik))
((0i, • • •, Qp)) is nonzero only if к = p and A((ir,..., ik)) = z((d15..., dk)). In
this case
(л(й> *   *7I(ik))((Qi, • • , 0л)) = X Qsi  • • Qsk-
H....= l-si 5 k I
VJ. d,j =ij
231
9.2 Idempotents
Proof By (1.5.7) and Lemma 9.24, we have
(7i(il) * • • • *	..., Qp)) = conck о ®	® n(ik)) ° <5fc((2b • • •, Qp))
= concfc ° (n(il) ® ® n(ik))^X (G/.) ® ® (Qik)^
= concfc^X 7i(il)((QIt)) ® • • • ® n(ik)((Q/k))^
where the summation is over all Ц,..., Ik as indicated in Lemma 9.24.
Observe that n(0 is the projection onto U(i), according to the direct sum
(9.2.1); hence, ti^R^ ..., Rt)) = 0 if I > 2 and if the Rj are Lie polynomials.
Moreover, л(1)(К) = 0 if R is homogeneous of degee / i. Thus, if some term
in the previous summation is nonzero, we must have V/ = 1,..., k, |/J = 1,
and for Ij={l], deg(Q() = ij. This implies k = p and A((i\,..., ik)) =
Ж-4))-
The previous discussion also shows that the last formula holds, because we
then have n(ij)((Qfj)) = Q,.	□
We say an algebra homomorphism </r. <□<£> -> <Q><4> is weight preserving if
for each i, (p(ti) is 0 or a homogeneous polynomial of degree i.
Lemma 9.26 Let Л = Г‘2"2.. .be a partition, P be a finely homogeneous
polynomial in Q<T>, of degree n( in each th Qt,... ,Qk be homogeneous Lie
polynomials such that 2((deg Qr,..., deg Qk)) = A. Then
C'(P)«ei,.. •, 2.» = E +<P(P),	<9.2.10)
<P
where the sum is over weight-preserving algebra homomorphisms Q<T> -»•
Q</1> such that cp(T) =^(Л), and which depend only on Qt,..., Qk.
Proof Suppose first that P = t(1... tik. Then £'(P) = C0 У(Р) = C(Mh... Mik) =
n(il) * • • • * n(ik), by (9.2.9) and (9.2.4), because KM = M„. By Lemma 9.25, we
thus have
c(PX(ei.- -.e»))=Ee.,- e».
where the summation is over all permutations .s^ ... sk of 1 ... к such that
V/, deg(Q;) = ij. Using inclusion-exclusion, this may be rewritten
sf П ...>,.)
\i>l	/
where the sum is over all £\, E2,... such that Et is a subset of {gy| deg Qj = i},
232
9 The Solomon descent algebra
and where (p is the homomorphism Q><P> -»• Q><4> such that <p(L) =
X<2e£. Q- Since such a <p is weight preserving, (9.2.10) follows in this case.
The general case follows by linearity.	□
Proof of Theorem 9.19 (a) Let Pe then P is a linear combination of
L, • •  4 with 2((1\,..., ifc)) = z.
Let Q e Uv; then Q is a linear combination of (Qb..., Qn), where the Qt
are homogeneous Lie polynomials in ^(4), and 2((deg Qi,..., deg Qn)) = v.
Since • • • tik) = я01)* • we deduce from Lemma 9.25 that
C'(P)(Q) = 0, unless X = v.
Now let Q any polynomial in Q<4>. By (9.2.1), Q = £v Qv, with Qv e Uv.
By what we have just seen
cw( e e,) = o,
\v^z /
hence,
c(p)° ял(2) = c'oej = ewe).
Thus, C'(P)° = C'(P), which implies that C'(P) e Г„лл.
(b) Let P e д again. Let 2 = Г‘2"2... and p = pr ... pt. Then P is
finely homogeneous of degree n( in each ti5 and also a linear combination of
(Pb...,?,) for some finely homogeneous Lie polynomials in Q<T> with
|Р;| = МЛ observe that if <p is any weight-preserving algebra homomorphism,
then <p((Pi,..., P,)) = (<z>(Pi),..., </>CP/)) and deg(</>(^)) = Щ1 = if more-
over </>(Л) c y7(7’), then each </>(Pf) is a homogeneous Lie polynomial of
degree pt. Thus, </>(Р)е Ц,.
Let Q be any polynomial in K<4>. By (a), we have C'(P)(Q) = C(P)° ttfiQ),
hence we may suppose that Q is a linear combination of (Qb ..., Qk) with
z((deg Qi,..., deg Qfc)) = z. This implies by Lemma 9.26 that C'(P)(Q) is a
linear combination of <p(P), hence is in Lf.
We deduce that ° C'(P)(Q) = C'C^XQ) for any Q in K<4>, hence
С(Р) = я/'(Р)еядГ„.
(c) By (a) and (b), С'(^,д) - ядГ„яА. Conversely, let f e лдГ„лА. Since £'
is surjective, and by Lemma 9.18, we have f = ^v>aC(Pva), with Pva e Wvx.
But we have seen that C(Pvat) e явГ„я„, and the sum (9.2.5) is direct. Thus,
f = С(Р^) e C'( И;д), and we deduce лдГ„ = £'( Иу.	□
The previous results may be interpreted in the descent algebra because
of Corollary 9.12.
Theorem 9.27 Let r: Q<T> = ©n>o he the linear isomorphism such
that T(tit... tik) = Лс, C = (ii,. .., ik). Define E- = т(Кх), where K, is given
by (9.2.3).
(i)	The elements Ex are, for |z| = n, orthogonal idempotents of sum 1 in the
algebra
9.3 Homomorphisms	233
(ii)	The linear mapping P i—► PEX, Q(Ajn -* 2<4>„ is the canonical projec-
tion corresponding to the direct sum
K<A\= © Ц.
|A|=r>
(iii)	For |Л| = \p\ = n, the space Ex is equal to т°у(И^д), where
is defined in (9.2.6) and у in (9.2.9). Its dimension is equal to the number of
multisets M of primitive necklaces such that /.(M) = Л and p(M) = p. In
particular, dim(£A Ef) = 1.
(iv)	The radical of^n is of dimension 2"-1 — p(n).
Similar results hold of course for the algebra (cf. Corollary 9.13).
9.3 HOMOMORPHISMS
Recall that A denotes the ring of symmetric functions over <□, and A„ the
subspace of homogeneous symmetric functions of degree n. Recall also that
A„ has another product, called inner product and which we denote by л,
defined by
(РлАл) л (pM/zM) = b^pjz^,	(9.3.1)
where zz = 1"'2"2... njnj!..., for z = 1"'2"2....
Theorem 9.28 Let |4| > 2. There exist, for |Л| = n, one-dimensional represen-
tations (р/. Гп ->Q such that:
(i)	The linear mapping q>„: Г„ -+ A„ defined by ipn = Jw=lt is an
homomorphism from Г„ onto A„ with inner tensor product, Ker (p is the radical
ofTn and (p„(nw) = pjzv
(ii)	For any polynomial P in Ux, |2| = n and any element f in Гп, one has
f(P) s Vl(f)P mod X	0.3.2)
n< i.
If p / 0, then (pfif) is completely determined by this equality.
Proof By Corollary 9.20, we have dim(nAr„nA) = 1. Since nz is idempotent,
we have = 0>ял. Define by = </>л(/)ял. Then is well
defined, because / 0 (indeed If / 0, because |Л| > 2), and linear. More-
over, by Corollary 9.22(ii), we have
Гп ©	ttp Г, 71^.
Ш = 1д1 =П
234	9 The Solomon descent algebra
Hence, each f in T„ has an expansion f =	with ядГплл. In
particular,	Then, if g is another element of Г„, we have
f9 X Ад £ 9av = E
a>v	Л > д > v
because the n/( are orthogonal idempotents. Thus
Qfe^i-V У, fi.n9nv1
А>д >v
and in particular (fg)u = f^g^.
This is rewritten
4h(f9)^ = (pJJ^xcp^g^x = <Pi(f)4h(9)7h.
This shows that <p; is an homomorphism, because лА / 0.
(i) The multiplication rule (9.3.1) shows that <p is an homomorphism. By
Corollaries 9.22 and 9.23, its kernel is the radical of T„. Since
we deduce </>д(лл) = 6^ and </>„(ял) = pjz^.
(ii) Since = 0 if 2 < p, we have, for P in If (thus лл(Р) = P),
f(P) = MP) = X nJn^P) = £
д	л. > д
= 9>л(/)^(Л+ X = 4>Af)P mod X Цх-
A > д	д < А
The last assertion follows from (9.2.1).	□
With the notation of the theorem, define
<p = © </>„: Г = © Г„ —> A = © A„.
л > О	n> 0	n > 0
Corollary 9.29 (|4| > 2). The linear mapping (p is an algebra homomorphism
from Г with convolution onto A with the usual product. One has
<P&M) = Pn/п, q>(q„) = hn.
We first prove the following lemma.
Lemma 9.30 Let PY,..., Pk be homogeneous Lie polynomials such that
2((deg P15..., deg Pk)) = L Then
p1...pk = (p1,...,pt)mo<i £ i/,.
д < Л
Proof (induction on к = /(2)). If /(A) = 1, there is nothing to prove.
Otherwise, the identity
P^. = P^. + [^,PJ
9.3 Homomorphisms
235
shows that the products Pa(1)... Pa(k), ct e Sk, differ from Pj ... Pk only by a
linear combination of products of homogeneous Lie polynomials Qi ... Qi
such that 2((deg ..., deg Qt)) < 2. By induction, these products are in
X/(< Цг Thus we have, for any a 6 Sk:
Pi Pk = Л(1> • • • ^a(fc) mod X Ч-
д < Л
Summing over all a in Sk, we get the lemma.	□
For a composition C = (i15 ..., ik), write тсс = n(h)* • • • * л(1к).
Proof of Corollary 9.29 Let C be a composition such that 2(C) = 2. Let
Q e Ц(. Then Lemma 9.25 implies that nc(2) = 0, unless p = 2. Hence, by
(9.3.2), we have </>M(nc) = 0 for p / 2.
Suppose now that Q e If and put 2 = Г’2"2.... Then Q is a linear
combination of (Q15..., Qk), with the notation of Lemma 9.25. By the
latter, we obtain that nc((2i,  • •, Qk)) is equal to a sum of s =	...
permutations of the product Qx ... Qk. Thus, by Lemma 9.30, we have
, Qk)) = s(Qn • • •, 2fc)mod£M<A Ц, and we conclude that
nc(Q) = SQ m°d £д<л Цх- This implies, by (9.3.2), that (p^(nc) = s. Thus, by
Theorem 9.28, we have
4>„(nc) = spx/zx = njnj ...	• • • 1"'2"2...
= (Р1/1)П,(Р2/2Г. • •
In particular, </>(я(в)) = p„/n and </>(n(i)) * • • • * n(ik)) = </>(n(il >)*•••* (p(it(M).
Since the nc span Г, we conclude that ip is multiplicative.
Now, by (9.2.2), we have
x'hf = log(l + tjx + • • • + tnxn + •••).
i> 1
Applying the homomorphism (p°f.	A, and noticing that C(L) = Чь
C(Mi) = 7i(i) by (9.2.4), we obtain
X x‘Pi/i = X xi(P(^i)) = log(\ + (p(qf)x + • • • + (p(q„)xn + •••)•
i > 1	i > 1
We deduce that
X 4>(qn)xn = exp(£x'p,/i) = X hnx”,
as is well known. Hence, cp(q„) = h„.	□
Corollary 9.31 (|Л| > n). The radical of T„ is generated by the elements
tic — 7tC' (respectively qc — qc ) with |C| = |C'| = n, 2(C) = 2(C')-
236
9 The Solomon descent algebra
Proof We have nc — itc- (respectively qc - qcfe Ker </>, under the stated
conditions, by Corollary 9.29. By Theorem 9.28(i), Ker is the radical of
Г„. Moreover, it is of dimension 2"-1 — p(n), by Corollary 9.23. Since the
elements nc (respectively qc) are linearly independent, there are 2" 1 — p(n)
linearly independent elements among the elements лс — itc> (respectively
Qc ~ Qc) satisfying the condition in the corollary.	□
All the results we have stated so far for Г„ have an interpretation in the
Solomon descent algebra S„, via the anti-isomorphism of Corollary 9.12. We
leave this translation to the reader, and now give a result which is better
formulated in terms of S„. For this, we call Lie monomial of type /. and
degree n a product Pr ... Pk of homogeneous Lie polynomials in ^f(A) such
that z((deg P15..., deg Pfc)) = z, and |Л| = deg(Pj ... Pk) = n.
Theorem 9.32 (|И| > n). An element x in QSn is in if and only if for each
Lie monomial Pr ... Pk of degree n, (PY ... Pk) x is a linear combination of
^a(l) • • • a G $k-
The direct part may be proved by using Lemma 9.25. However, we give
another method, interesting in itself.
Lemma 9.33 Let C = (ji,   , ji) be a composition of n and Px ... Pk be a
Lie monomial of degree n. Then
Р„)ЛС = qc(P, . .. P.) = X Ps, ... PS1,
where the summation is over all partitions {1,..., k} = u • • • u S1 such that
deg PSi = j(, with Ps = P|ieS Pi (product in the natural order of S).
An example of such a formula is the following (where subscripts indicate
degrees):
(?1^262)^23 = P2P1Q2 + Q2P1 ^2-
Proof The first equality is (9.1.5). Now, for С = (jb..., J,), we have
<7с(Л . .. Pk) = (qh * • • • * qjl)(Pl ... Pk)	(by (9.1.2)
= cone, - (qh ® • • • ® qjt) ° ^(Pi    Pk). by (1.5.7).
Now, if P is a Lie polynomial, then by (1.5.6)
3,(Pi) = Pi ® 1 ®	® 1 + 1 ® Pi ® • • • ® 1 + • • • + 1 ® 1 ®	® Pf
This shows that
<5,(P,... P,) =	. <5,(PJ = X Ps, ®' • -® Ps„
9.3 Homomorphisms	237
where the sum is over all decompositions {1,..., k} = SY и • • • и Sh with
Ps = FLes ?t- Applying cone, ° (<?j, ®	® q71), we get the lemma. □
Proof of Theorem 9.32 Since the Ac, for |C| = n, generate linearly E„, the
only if part of the theorem follows from Lemma 9.33.
Conversely, let x be as in the theorem. We may suppose that A contains
{1,..., n}, and we identify a permutation S„ with the corresponding word
in A*.
Let C be a composition of n, C = (c15..., ck), and factorize the word
12 ... n into Uj ... uk with |u,-| = c(. By (3.2.1), is a homogeneous Lie
polynomial of degree cf. By hypothesis, we have
(^C1)(M1) • • • ^(C(<)(Wfc))x E
aeSk
for some coefficient fxC in <□. Since the q, commute with substitution of
letters, the same holds for the qc by Lemma 1.11; the latter span Г, hence
each n(i) commutes with substitution of letters. Moreover, so does the
mapping P и-> Px. Hence, if v1,...,vk are words of respective lengths
c15..., cfc, we obtain
Wl) . . . 7lM(vk))x = X	. .7l(c,lk>)(Vxlk)).
aeSk
Now, observe that by (1.5.7) and Proposition 1.8
Ko*' • •* я(Ск))(12 ... n) = X (12 ... n, Vi ш- • -ш ^Х^)... 7iM(vk).
kJ =g
Applying x on the right, we obtain
[(я(С))*-• •*я(Ск))(12 ... n)]x
= X E (12---n, Ui Ш-•mvJ/JiCn(cJt,a(1))... n(f,lk))(ra(fc))
|i’,| -c, aeSi<
= E fa.C E (12 ••• П, Vx(l) Ш • • • Ш VxW)n(c,a/Vx(l)) . . .7CM(vx(k))
«eSk |r3(1|| =c3(,)
= E	••*Я^к,))(12...И).
aeSi<
Let Ic denote the element of corresponding to л(С1) * • • • * n{Cki through the
linear isomorphism of Corollary 9.12. Then Ic =	 •* W12...4
and we deduce from (8.1.4) that Icx = Y^xfx.c^ca (product in KS„), where
Cot = (ca(n,..., ca(fc)). Since the Ic span and since has a unit element,
we deduce that x is in S„.	□
Suppose that the alphabet A is infinite. Then, by Corollary 9.14, the
«□-algebra Г (under convolution) is freely generated by the elements qn, n > 1.
We may therefore define for each s > 1 a derivation of the convolution
= I
238	9 The Solomon descent algebra
algebra Г by
Qn-S'
where qk = 0 if к < 0.
Theorem 9.34 For each s > I, the derivation f of the convolution algebra Г
is also a homomorphism of the algebra Г under composition, sending r„ + s into
Г„. Moreover, for each partition /.
ifs is a part of 2,
0 otherwise.
Proof (a) We have to show that £s(qH °qK) = £S(</H)0 for any pseudo-
compositions H, K. Since cs is a derivation, we have for H = (hY,..., hk),
L = (ll,...,lp):
UQh) =	* • • • *	= X
1 <i<k
s <h,
where Ht is obtained from H by replacing h, by h( — s. Similarly,
SM = X
i <j<p
S <lj
where Lj is obtained from L by replacing lj by Ij — s. Thus, by Theorem 9.2,
= E E«««,».	<9-3-3’
hi,l} >s
where the second sum is extended to all matrices Afy whose row sum is
and column sum is Lj. On the other hand, we have by Theorem 9.2
^s(Qho(1l) =	,
where the sum is extended to all matrices M = (mf) whose row sum is H
and column sum is L. Thus,
£sG?h0#l) = X X	(9.3.4)
M m,j > s
where the matrix M(j is obtained from M by replacing the entry mtj by mtJ — s.
Note that has row sum Ц and column sum Lj, so that the sum (9.3.4)
is contained in the sum (9.3.3). Conversely, if a matrix Мц how row sum Hi
and column sum Lj, then by adding s to its (i,j)-entry, we obtain a matrix
of row sum H and column sum L. Hence, the reverse inclusion also holds,
and (9.3.3) is equal to (9.3.4).
9.3 Homomorphisms	239
(b) Since A is infinite, the algebra homomorphism £Q<T> -* Г (con-
volution), with C(t„) = q„, is an isomorphism (Corollary 9.14). Define a
derivation ds of Q><T> by ds(tn) = tn_s, with t0 = 1, tk = 0 if к < 0. Then, by
Theorem 9.16, all we have to show is that
ds(Ki)
ifsisapart ofA,
otherwise.
(9.3.5)
Extend ds to the variables x, x15 x2,..., used in (9.2.2) and (9.2.3), by
ds(x) = ds(xi) = ds(x2) =    = (). Then
ds( E	) = E ds(ti)x‘ = X 11-*х1 = xs X Ьх1.
\i > O	/	i>0	i>0	i>0
Hence, X tix' and its image under ds commute. Observe that if у and ds(y)
commute then
ds(y”) = nyn 4(y).
Thus, by (9.2.2)
X xnds(Mn) = dA X x"M„ = dA log! 1 + X *<х‘
n > 1	\л >1	/	\	\	i > 1
\k > 1 К
with S = X;> i h*1- Now, the latter sum is
X —y— ds(Sk) = X -7- kS'-'dtf) = (1 + SyXCS)
k>i к	k> i к
= (i + s)~4(i + s)
(\ -1 /	\
X ttx‘I xsl X hx* ] = xs.
i>0	/	\i>0	/
This shows that
X xU(JW,) = x-,
n > 1
which implies
W) =
Now, let U = X«>i х(М(. Then
ds(U) = X
240	9 The Solomon descent algebra
so that U and its image under ds commute. Thus,
Js(exp((/)) = J E 7/''*)= E
\k > О к'.	/ к > 0 KI
= X /~кик~Ч5(и) = exp(U)xs.
fc>o kl
With (9.2.3), we deduce
E x^ds(Kx) = ds\Y = ds(exp((7)) = (£ x2K^xs = X x^sK^.
>.	\ Л	/	\Л/д
Thus (9.3.5) holds.	□
Further properties of the mapping cs are given in the following result.
Theorem 9.35 The mapping cs is surjective. For s = 1, 2, it admits a right
inverse, which sends T„ into Tn+s, and which is a homomorphism for the algebra
Г under composition.
In particular, each algebra Tn is embedded in Г„+1.
We begin by a lemma. Denote by c/ca the derivation of Q<4> (concatena-
tion algebra), with A = {a, b,...}, such that d/да sends a to 1 and the other
letters to 0.
Lemma 9.36 Let E denote the subspace of generated by the poly-
nomials
(a,P2,...,Pn),	(9.3.6)
where n > 1 and where P2,..., Pn are Lie polynomials in ^(A). Then the
restriction of d/да to E is a linear isomorphism onto
Proof Observe that the kernel of c/ca is a subalgebra of Q<Acontaining
b, c,..., and closed under the operation P [а, Р]. Hence, each finely
homogeneous Lie polynomial, which is not a scalar multiple of a, is in this
kernel.
Observe also that (1, P2,..., P„) = (P2,..., P„), and that in the definition
of E, we may suppose that each P{ in (9.3.6) is finely homogeneous, and is
equal to a if it is a scalar multiple of a. Under these conditions, let j > 0
denote the number of Pt in (9.3.6) equal to a. Then, by symmetry and
multilinearity of the operator (,....,), and by the previous remarks, we
9.3 Homomorphisms
obtain
241
3
— (a, P2,..., P„) = (j + l)(P2,...,f>,).
da
Thus, d/da is surjective, because Q<A> is generated by the polynomials
(P2,...,£„), by (9.2.1). A counting argument, as in Lemma 8.22, shows that
d/da | E is injective.	□
Proof of Theorem 9.35 (a) The convolution algebra Г is isomorphic with
<□<£>, via the isomorphism <□<£> Г. Let ds be the derivation of Q<£>
sending t„ onto t„_s, with t0 = 1, and tk = 0 if к < 0. We thus have
Let E be the subspace of Q<£> generated by the polynomials
(ts, P2,..., P„), where the Pt are Lie polynomials in ^f(T). By Lemma 9.36
the restriction of d/dts to £ is a linear isomorphism £	<□<£>.
With у the algebra automorphism of <□<£> defined by (9.2.9), let
d's = 7-1 °ds°y.
Then d's is a derivation of Q<T>, and we have d'^t^ = y-1 ° ds(Mn). In the
proof of Theorem 9.34, we have seen that ds(Mn) = dsn. Thus d's(t„) = dns, and
we conclude that d's = d/dts. Hence, d's\E is a linear isomorphism £ ->• <□<£>.
This implies that ds\y(E) is a linear isomorphism y(£) -► Q<T>. We conclude
that £s\£°y(E) is a linear isomorphism £ ° y(£) Г, and in particular £s is
surjective.
(b) Observe that a finely homogeneous Lie polynomial P in Q<T> of
weight 1 (respectively 2) must be a scalar multiple of (respectively t2).
Thus, for s = 1, 2, the subspace £ is equal to
©
seA, sejt
by definition (9.2.6) of W2 Hence, we obtain by Theorem 9.19
C"T(£) = ('(£)= © явГЛ.
se Л. se д
Since the лл are orthogonal idempotents, the subspace C (£) is therefore a
subalgebra of Г under composition, with neutral element	We have
seen that
es|C'(£): C'(£)-r
in a linear isomorphism. Since cs is an homomorphism for composition by
Theorem 9.34, £S\C(E) is an isomorphism of algebras, and £s has a right
inverse.	□
242
9 The Solomon descent algebra
9.4 QUASISYMMETRIC FUNCTIONS AND ENUMERATION
OF PERMUTATIONS
Let X be a totally ordered infinite set, which will serve as an alphabet, and
also as a set of commuting variables. A formal power series F in Z[[X]] is
called a quasisymmetric function if for any x15..., x„, yY,..., yn in X, with
Xi < • • • < x„, yr <  • • < y„, and any positive integers kl,..., kn, the coeffi-
cients of x*‘... x*" and y?' ... y*H in F are equal. We denote by Qsym the
ring of quasisymmetric functions, and by Qsymn the Z-module of homo-
geneous quasisymmetric functions of degree n.
If C = (i15..., ik) is a composition of n, we define Mc by
Мс= X X? •••4.	(9.4.1)
Xi < ’ • • < xk
Clearly, the Mc form a basis of Qsymn, for |C| = n. Define Fc by
Fc= Z MD,
D>C
where D > C denotes that the composition D is finer than C, i.e. S(D) 3 S(C),
where S(C) = {i\, + i2,..., + • • • + ik-i} is the subset of {1,..., n — 1}
associated to C. By inclusion-exclusion, we have
Mc= £ (- 1)“di’w’Fd.	(9.4.2)
D>C
where /(C) is the number of parts of C. This shows that the Fc are a basis
of Qsym„, for |C| = n. Let С = (ц,..., ik) be a composition of n, and
5 = S(C). The definition of Fc shows that
Fc = £ x, ... x„	(9.4.3)
xieX
where the summation is subject to the condition: for each i = 1,..., n — 1,
x, < x1 +15 and xf < xi+ i if i 6 S. For example,
Л1.2) = M1.2) + A/d.i.D = S xy2 +	£ xyz= X XiX2X3,
X<y	X<y<Z	X1<X2<*3
and the set associated to the composition (1, 2) is {!}.
Let У be a second infinite totally ordered alphabet, disjoint from X. We
may consider the set
Z = {xy|x 6 X, у 6 Y}	(9.4.4)
as an alphabet, totally ordered by
xy < x'y' if either x < x' or x = x' and у < у'.	(9.4.5)
Thus, if FeZ[[X]] is a quasisymmetric function, then it makes sense to
9.4 Quasisymmetric functions and enumeration of permutations	243
consider F(xy) e Z[[Z]]. There is a canonical algebra homomorphism
u У]]. We may without ambiguity identify F(xy) with its
image under this homomorphism. Indeed, the latter maps homogeneous
components of degree n on homogeneous components of degree 2n; further-
more, if F(x) is a homogeneous quasisymmetric function of degree n, then
for fixed x0 in X, we have F(xy) = x^F(y) + terms of lower degree in x0.
Recall that C(cr) denotes the descent composition of о (see Section 9.1).
Theorem 9.37 Let о be a permutation in S„. Then
FC(a)(xy) = X FCM(x)FC(P)(y).
a = fla
Recall that we have defined a basis (Ds) of the Solomon descent algebra En,
indexed by subsets of {1,..., n — 1}; see Section 9.1. The theorem shows
again that is a subalgebra of <Q>S„: indeed, since the series Fc(x)FD(y) are
linearly independent, the number of pairs (a, ft) e S„ x S„ with C(a) = C',
C(f) = C" and fa = о depends only on C = C(a), C', and C". We thus have
the following result.
Corollary 9.38 Let S, S', S" be subsets of (1,..., n — 1} and С, С, C" the
corresponding compositions. Then the coefficient of Ds in the product Ds. Ds,
(expanded in the Ds basis) is equal to the coefficient of Fc-(x)Fc--(y) in Fc(xy)
(expanded in the Fc(x)FC"(y) basis).
For the proof of Theorem 9.37 we shall need a couple of lemmas. Recall
that the standard permutation a = st(w) of a word w = Xj ... x„ is defined
by
<r(0 < </)	(*> < Xj) or (Xi = Xj and i <j).	(9.4.6)
Denote by ev(w) the evaluation of w, i.e. the monomial Xj ... x„ in Z[[X]].
Lemma 9.39 Let a be a permutation in Sn. Then
FC(a-')= E W(w).
St(W) = <7
Proof We have by (9.4.3), with C = C(cr-1) and S = S(C):
Fc =
where the sum is over all increasing words .. t„ of length n in X* such
that for each i in S, < ti+ P By putting xf = ta(i), we deduce
Fc = X w(xi • • • x„)’
244	9 The Solomon descent algebra
where the sum is over all words such that
<	< xff-1(ll) and V/ceS, x„-i(k) < x„-i(k+1).	(9.4.7)
We show that (9.4.7) is equivalent to the condition that a = st(xr ... x„),
which will prove the lemma. Since C is the descent composition of a"1, S is
the descent set of a-1, and we have
keSoo'^k) > o~fk + 1).	(9.4.8)
This means that к is at the right of к + 1 in the word cr(l)... cr(n).
Suppose that (9.4.7) holds. We show that o(i) < a(j) implies the right-hand
side of the equivalence (9.4.6): this will imply (9.4.6), because each side defines
a total order on {1,..., n}, and thus о = st(w). So let cr(i) < o(j). By the
first condition in (9.4.7), we have
— Xa - 1 (<r(i)+ 1) < ' ' ‘	- i(<r( j) - 1} —	%j-
By the second condition in (9.4.7), we deduce that, if x, = x7, then
S n {o-(i), o-(i) + 1,..., <t(j) - 1} = 0.
Thus, by (9.4.8), a(i) is at the left of o(j) in the word cr(l)... cr(n), showing
that i < j. Hence, the right-hand side of (9.4.6) holds.
Conversely, suppose that a = st(xY ... x„), i.e. that (9.4.6) holds. We show
that (9.4.7) holds. We have i < i + 1, hence aafi)<aa *(i + 1). By (9.4.6),
this is equivalent to
(x.x-ni) < xff-1(i + 1)) or (xff-1(i) = xff-1(i + 1) and a-1(i) < o~1(i + 1)).
Because of (9.4.8), we deduce that (9.4.7) holds.	□
Lemma 9.40 Let и = xt ... x„ e X*, v = y1...y„eY* and w(u,v) =
(Х1У1)... (x„y„) e Z*. Then st(w) = st(ust(v)~l)st(v), where ust(v)~r denotes
the right action of the permutation st(r)”1 on the word u.
Proof We have
st(w)(i) < st(w)(j)
(х.У.-	< Xjyj) or (x.y,- = x}y} and i <j)	(by (9.4.6)
(Xj <	x^ or (x; = Xj and y, < yj) or (xt- = x7 and yt	= y7 and i <j) by (9.4.5)
(X,- <	Xj) or (Xj = Xj and (y, < y7 or (y, = y7 and i <	'Ш
(xf <	Xj) or (x(. = Xj and st(v)(i) < st(y)(J))	by (9.4.6)
9.4 Quasisymmetric functions and enumeration of permutations	245
Denote ust(v)-1 =tr...tn, hence xt = tst(v)(i). Then the previous condition
is equivalent to
(^st(v)ti)	or Qst(v)(i)	and st(u)(i) < st(u)(j))
О st(ti ... tn)(st(y)(i)) < st(ti ... t„)(st(v)(j)) by (9.4.6)
оst(ust(v)~ 1)st(y)(i) < st(ust(y)~ 1)st(v)(j).
This shows that st(yv) = st(ust(v)~ r)st(v).	□
Proof of Theorem 9.37 By Lemma 9.39,
Fc(a)(xy) = X ev(wF
weZ*
st(w) = a ~ 1
where Z is defined in (9.4.4). For u, v, w as Lemma 9.40, write w = w(w, v),
and let ev(w(u, v)) = ev(u)ev(v). Then
FC(a)(xy) = X ev(yv(u, v))
ueX* vet*
st(w(u, i»)) = a ~ 1
= X ev(u)ev(y)
st(ust(v) " 1 )st(v) = a ~ 1
= X X ev(ust(v)~ ^eviv)
a = fix st(ust(v) ~ 1) = a  1
st(v) = fl- >
= X X ev(v>) X evfuf'1)
a = fix st(v) =fl~ 1 stfufl ~ 1) = x ~ 1
= x ( x ^(y)Y x ev(u'))
a = f}x	=	/ \st(u') = x ~/
— X FC(p)(y)FC(ai)(x),
a = fix
by Lemma 9.39	□
Let П be a set of permutations, i.e. П c U">o We call quasisymmetric
generating function of П the quasisymmetric function
I Fc,.,-
леП
We compute below this generating function for special sets П, and it will
turn out that it is actually symmetric and equal to the characteristic of a
representation of the symmetric group.
Recall that we have defined subspaces If of K<4> in Section 8.5. The
evaluation of a multiset of primitive necklaces has been defined in Section
7.5. For a composition C = (i15..., ik), denote by hc the product П1 < j<k hi}
of the corresponding complete symmetric functions.
246	9 The Solomon descent algebra
Theorem 9.41 (i) Let л be a partition of n. Then the quasisymmetric
generating function Pk of the set of permutations of cycle type 1 is equal to the
generating function of the set of multisets of primitive necklaces of cycle type
z, and also to the characteristic of the representation of S„ on the multilinear
part of If.
(ii) Let C be a composition of n. Then the quasisymmetric generating function
Sc of the set of permutations whose inverse has descent composition C is equal to
Sc = E (-1)*'-^,,	(9.4.9)
D<C
and is the characteristic of some representation of S„.
Proof (i) The bijection of Theorem 7.20 maps bijectively words w whose
standard permutation has cycle type A onto multisets of primitive necklaces
of cycle type z. Since this bijection preserves the evaluation and since the
inverse of a permutation has the same cycle type as it, we obtain the first
assertion as a consequence of Lemma 9.39.
The polynomials Qw introduced in the proof of Lemma 8.22 have the same
partial degrees as w. Hence, by Theorem 8.1, the characteristic of the
representation of Sn on the multilinear part of If is equal to the sum
of the evaluations of the words w whose decomposition into Hall words
w = iq ... hk satisfies: z((/q|, • • •, |Л*0) = z. Since Hall words are in evaluation-
preserving bijection with primitive necklaces (Corollary 7.5), this sum is
equal to P,.
(ii) Let С = {q,..., ik} and S = {q, f + i2,...	+ • • • + ik_ J. We
claim that C(st(w)) < C is equivalent to
Vi£S, x, < xi+1,
where w = xt ... x„. Indeed, C(st(w)) < C is equivalent to D(st(w)) £ s, i.e. to
Vi ф S, st(w)(i) < st(w)(i +1),
and this condition is, by (9.4.6), equivalent to
Vi ф S, Xi < xi+1 or Xi = xi+ j and i < i + 1.
Hence, the claim follows. It implies that
E ev(w) = hil...hik = hc.
C(st(w)) < C
This implies that
hc= E E E ev^ = E E Fc(a-q= E sd-
D <C C(a) = D st(w) = a	D<C C(a) = D	D<C
by Lemma 9.39
Hence, we obtain (9.4.9) by inclusion-exclusion.
9.4 Quasisymmetric functions and enumeration of permutations	247
The same argument shows that Sc is the sum of the evaluations of the
words w = *! ... xn satisfying the condition:
Vi ф S, x( < xi+1 and Vi e S, x( > xi+1.	(9.4.10)
This shows that Sc is the skew Schur function corresponding to the
skew-hook whose lengths of rows are determined by C (from top to bottom,
in the French way of depicting tableaux); in particular, Sc is a sum of Schur
functions, and hence is the characteristic of a representation of S„; see
Macdonald (1979).	□
Recall that the space Л of symmetric functions has a scalar product <, > for
which the bases (hf) and (шл) of complete and monomial symmetric
functions are orthogonal to each another. Observe that each symmetric
function is quasisymmetric. The next result gives its expansion in the Fc basis.
Theorem 9.42 Let g be a symmetric function. Then
c
Proof We have
4 = E	= E <<Л M E MC
л	л	л(С) = л
by (9.4.1)
X <9.Лл>Мс = Х<»,Лс>М<
л(С) = Л	С
because z(C) = z implies /ic = /iA.
Hence by (9.4.2)
0 = E <0’ hc> X (- W)FD
C	D>C
= Y.Ffe. L (-l)'<D,-"ohc) = LF„<</,Sd>
D \ C < D	ID
by (9.4.9).
□
Corollary 9.43 If the quasisymmetric generating function g of a set П of
permutations is symmetric, then for any composition C, the number of
permutations in П whose descent composition is C is equal to the scalar product
((J, Sc>.
Proof Let the number in question be ac. Then g = xcFc. Since the Fc
are linearly independent, we obtain from Theorem 9.42 that ac = <3, Sc>. U
248	9 The Solomon descent algebra
Corollary 9.44 Let C be a composition and A a partition. The number of
permutations of cycle type A whose descent composition is C is equal to the
scalar product Sc).
Corollary 9.45 Let C, D be two compositions. The number of permutations
which have descent composition C and whose inverse has descent composition
D is equal to the scalar product <SC, SD).
Both corollaries are an immediate consequence of Theorem 9.41 and of
Corollary 9.43.
9.5 APPENDIX
9.5.1 Graded bialgebras
The projections q„ and the convolution subalgebra Г generated by them may
be defined in each graded bialgebra. Theorem 9.2 and Corollary 9.11 are
valid if the component of degree 0 of the bialgebra is К and if it is
cocommutative (the latter property is used in Lemma 9.10). When the
bialgebra is commutative instead, Theorem 9.2 and Corollary 9.11 are
still valid if one replaces ‘row’ by ‘column’ in the definition of the
pseudocomposition associated to a matrix. The proofs of these generaliza-
tions follow closely the proof given here.
The previous remarks where suggested by an idea of A. Joyal.
9.5.2 Mackey formula
Let C = (i15..., ik) be a composition of n, f о • • - о Ik the partition of
{1,..., n} into consecutive intervals such that the length of Is is is, and Sc
the subgroup of S„ consisting of the permutations which keep this partition
invariant. Such a subgroup is called a Young subgroup. Denote by Xc the
set of permutations о such that cr| 7S is increasing, for any s. In other words,
Xc is the set of permutations whose descent set is contained in S(C). Then
Xc is a set of representatives of the right cosets aSc.
Let D = (jji) be another composition of n, f и  • - о J, the cor-
responding partition, SD the corresponding Young subgroup and XD the
corresponding set of representatives of the right cosets.
Then XCD = Xc1 n XD is a set of representatives of the double cosets
ScoSD (see Bourbaki 1981a, Chapter 4, Exercise 1.3). For each о in XCD,
a~lSco n SD is a Young subgroup SE, and the multiplication table of the
basis elements Ac of the Solomon descent algebra Sn is given by

(9.5.1)
9.5 Appendix
249
where the summation is over all these Young subgroups SE; this is established
in the case of a finite Coxeter group by Solomon (1976); recall that the
symmetric group is such a group.
The link with the multiplication table given here (Theorem 9.2 together
with Corollary 9.12) is given by the following result (James and Kerber 1981,
Section 1.3): the mapping
(|JS n <l(J()|)i <s<k.
is a bijection from XCD onto the set of к by I matrices over N of row sum
C and column sum D (actually, the matrix above depends only on the double
coset of cr). The precise link with (9.5.1) is that, if E denotes the composition
obtained by reading the matrix above column by column, and omitting zeros,
then
SE = (TlSca n SD.
\
Recall that the algebra Sn and Tn are anti-isomorphic, via the linear mapping
Ac -> qc (Corollary 9.12). Using this bijection and the homomorphism <pn
of Theorem 9.28, (9.5.1) implies by Corollary 9.29 that the inner product of
the symmetric functions h, and is given by
Ал л *„ = £*,,,	(9.5.2)
where the sum is over all matrices over N whose row sum is z and column
sum is p, with v the partition determined by the nonzero entries of this matrix.
Recall that h; is the characteristic of the representation of Sn(|z| = n)
induced from the trivial representation of the Young subgroup Sx (Macdonald
1979, Section 1.7), and that the inner tensor product л of the symmetric
functions corresponds to the inner tensor product of representations. Trans-
lated in this way, (9.5.2) may be established directly (James and Kerber 1981,
Section 2.9). On the other hand, in the language of double cosets, (9.5.2) is
a special case of the Mackey formula (Curtis and Reiner 1962, 44.3).
9.5.3 Idempotents of the canonical projections
Denote by En k(l < к < n) the following element of Sn:
E.,t = L E,.
|л|=и, iw = k
Then the En k are for к = 1,..., n orthogonal idempotents of S„, whose sum
is the identity (see Theorem 9.27). They correspond to the canonical
projections of the free associative algebra К (A), corresponding to the direct
sum
K<4>= © Uk
k>0
250	9 The Solomon descent algebra
(see Section 3.2), in the following sense: if P is a homogeneous polynomial
of degree n, then its image under the canonical projection К (A) -> Uk is
PEn k (right action). This follows from Theorem 9.27, because
C4= ® и,.
lW = k
If we replace x,- by txl in the definition (9.2.3) of Kx, we obtain
where s(m, k) denotes the Stirling number of the first kind (see Comtet 1970,
p. 47). Thus, we obtain, for any k and n,
1(Л) = к	т>к	m.
Щ = n	ii + • • • + ik = л
Applying the mapping r of Theorem 9.27, we obtain an expression for En к
in the group algebra of Sn:
E,.b= E	£ Ac.	(9.5.3)
m>k	ml l(C) = k
|C| = n
The idempotents En k were introduced and computed by Reutenauer (1986b),
in connection with the canonical projections, the Baker-Campbell-
Hausdorff formula and certain representations of the symmetric group.
The idempotents En t appear implicitly in Solomon (1968b), and were also
considered by Helmstetter (1989). Formula (9.5.3) is given by Garsia (1990),
who also gives the following generating series: let x be a variable; denote
(x) f" = x(x + 1)... (x + n - 1),
and let d(o) be the number of descents of ere S„. Then
Ё xkEn,k=~ E (x-d(a))]no.
k= 1	П! aeSn
9.5 Appendix	251
This may be deduced from (9.5.3), by using the well-known formula
(*)?"= X s(n,k)xk,	(9.5.4)
к = 1
(see Comtet 1970, p. 48). For details, see Garsia (1990).
A remarkable fact is that these idempotents appear, up to the auto-
morphism of the algebra <Q>S„ which multiplies each permutation by its sign,
in the quite different context of Hochschild homology. We give, without
proofs some of the results (the reader has to apply everywhere the previous
automorphism in order to recover the original results). Gerstenhaber and
Schack (1987) define an element s„ of <Q>S„ by
n -1
sn = X
i= 1
They show that the minimal polynomial of s„ in QS„ has the n distinct roots
2‘ — 2, 1 < i < n. From this they deduce that the n elements
e.(k) = ( П 2‘ - 2') ‘ П (s. - 2'' + 2), 1 < к < n,
\i^k	/ i ^k
are orthogonal idempotents whose sum is 1. They show that
e„(2) + • • • 4- en(ri) is the Barr idempotent (Barr 1968). Loday (1989) shows
that the idempotents e„(k) satisfy the following equation:
(-1)‘-	= ke„m + кге„(2) +  •  + k’eM,
where
and
/^(-l)*’1 X Ds.
|S| =*- 1
He shows that the Я* satisfy
Garsia and Reutenauer (1989) show that en(k) = En k. See also Patras (1990,
1991) for the study of these idempotents.
9.5.4 Multiplication rule
Let s be an element of the free magma M(T), and denote also by s its
canonical image in the free Lie algebra Denote by A(s) the partition
Aj ... xk, if tA1,..., tAk are the leaves of s, with multiplicities. As an example,
252	9 The Solomon descent algebra
z([[? p hl, G33) = 3211. Define the weight |s| of s to be |A(s)|. Now, let
s15..., st be several elements in M(T) and let
S = (s15..., s^.
Observe that if g(S) = A((|sj|,.. .|s,|)) = g and A(S) = A(Sj ... sf) = Л, then
S 6 M. We give a multiplication rule for the elements £'($) in Г. Let
R = (r15..., rk) another such element. Define R' = (r'15..., r'k), where each r'-
is in M(T), with
T = {/1, tp ..., t2, t2,   -, tn, t'n,...},
in such a way that the letters in R' are distinct, that T' is the set of letters
appearing in R' and that R is obtained from R' by forgetting the primes. Then
where the summation is extended to all weight-preserving magma homo-
morphisms <p: M(T) -> M(T) such that for i = 1,..., I, there is exactly one
t in T' with <p(t) = s,. As an example,
S = (Sp S2, $3, s4) = ([tp t2], t3, [t2, [t2, 11 ]], [t2, L^2’ f 133),
Я = (G’ E^ ^3, t5).
Then
R' = (t3, [t'3, t53, t'5).
There are four possible homomorphisms <p, given by
(pt3 = Sj or s2, (pt'3 = s2 or s15 <pts = s3 or s4, cpt'5 = s4 or s3.
Thus £'(R)£'(S) is the image under £' of
2([tp ^гЗ, Е*з> E^2’ G333, E^2’ E^2’ G33)
+ 2(t3, EEg, ^3, E^2’ E^2, G333, E^2’ E^2’ ^133)-
The proof of this multiplication rule may be done by using Lemma 9.25 and
by adapting the proof of Lemma 9.26.
Observe that the elements S are not linearly independent, but one can use
a Hall set in order to obtain a basis (cf. Lemma 9.18); in this case, one has
to apply the algorithm of Section 4.2 after the multiplication rule.
9.5.5 Another calculation of the multiplities
By Foulkes (1980), Zelevinsky (1981), or Gessel (1984), the scalar product
<sA, Sc~) is equal to the number of standard tableaux T of shape A, whose
9.5 Appendix
253
descent composition is C (the descent composition of T is the composition
associated to the descent set of T; see Section 8.3). Since the Schur functions
form an orthonormal basis, this remark implies that Corollary 8.10 is
equivalent to
— X Sc,
maj(C) = i mod n
(9-5-5)
where maj(C) = i (ci + ''" + ci), for C = (c15..., ck), and where i is a
fixed integer relatively prime to n.
A direct proof of this formula is due to Gessel and goes as follows.
Let
Gn(q) = X Scqm^c\
|C|=n
Hence, by (9.4.10), we have
Gn = X ev(x^ ... x„)qmaj<X1 Xn\
where the sum is over all words ... xn on X and where majC^ ... x„) is
the sum of the elements of the descent set of Xj ... x„. By a result due to
MacMahon, the number of words in {1,..., r} with evaluation Г'2П2... гПг
and major index к is the coefficient of qk in the q-multinomial coefficient
ni + n2 + ''' + nr
n15 n2,..., nr
(9.5.6)
see Stanley (1986, Proposition 1.3.17). Thus,
where we put X = {x15 x2,..Let to be some primitive Jth root of unity,
for some d dividing n. Then, a result of Gloria Olive (1965) implies that the
q-multinomial coefficient (9.5.6), evaluated at q = to is zero, unless each n, is
divisible by d, and in this case it is equal to the ordinary multinomial
coefficient
(n/d \
njd, n2/d,..., nr/dj
Thus,
G» =	£ f n/d
mi + m2 + ' ' ' =n/d ^2’  • /
= (xd + xd+---)"/d = pndld.	(9.5.7)
254
9 The Solomon descent algebra
Now, if f(q) = fkqk is a polynomial in q, then
Лп= z A = ‘(/(i) + m'i) + m_2)+ " + ci"‘iW’”*1)),
к = i mod n И
where C is a primitive nth root of unity.
Thus, by (9.5.7),
I n- 1
/7	_ У rkingcd(k,n)
'-6i[i]-Д'» Pn/gcd(k,n)
П k = O
gcd(r,d)= 1
If a) is a primitive Jth root of unity, then the Ramanujan sum
Z
0 < r < d
gcd(r, d)= 1
is equal to n(d). Hence,
G„[.i = 1 E PdW) =
П d I n
which proves (9.5.5). A by-product of this proof, or of Theorem 8.9, is that
for a symmetric function f of degree n, the polynomial
g(q) = Жq2,  )(i -4)0 -q2) 0 -qn)
satisfies the following property: the sum of the coefficients in g of the powers
qk with к = i mod n depends only on gcd(i. n). For this and a similar proof
of (9.5.5), see Desarmenien (1989).
9.6 NOTES
Theorem 9.2 is reminiscent of eqn (4.5) of Garsia and Remmel (1985): they
give a formula for the product 0(Ac)0(AD), in terms of matrices with given
row and column sums, showing directly that is a subalgebra of QSn. The
fact that £„ is a subalgebra of <Q>S„ (Corollary 9.12) was first proved by
Solomon (1976), in the more general case of finite Coxeter groups; his result
was generalized by Moszkowski (1989). In the case of the symmetric group,
besides the proof of Garsia and Remmel, there is another proof due to Gessel
(1984, Corollary 12; see also the remark following Theorem 9.37 in this
chapter).
All the results of Section 9.2, together with Theorem 9.28, Corollary 9.29,
Lemma 9.30, Theorem 9.32, and Lemma 9.33 are from Garsia and Reutenauer
9.6 Notes	255
(1989). Corollary 9.31 is already in Solomon (1976). Theorems 9.34 and 9.35
are from Bergeron et al. (1992b). Theorem 9.37 and Corollary 9.38 are due
to Gessel (1984), who also proved a result equivalent to Lemma 9.40. The
idea behind Lemma 9.39 goes back to MacMahon and was extensively
developed by Stanley (1972). Theorem 9.4 l(i) appears essentially in Bergeron
et al. (1988). The symmetric functions Sc where considered by MacMahon
(1960), and the corresponding representations by Foulkes (1980) and
Solomon (1968a). Theorem 9.42 and Corollary 9.45 are due to Gessel (1984),
and Corollaries 9.43 and 9.44 are from Gessel and Reutenauer (1992). Many
results concerning the Solomon algebra have been extended to other finite
Coxeter groups. The case of the hyperoctahedral group Bn has been
intensively studied; see N. Bergeron (1991,1992) and Bergeron and Bergeron
(1992a, 1992b). A global construction of the idempotents of the descent
algebra is made for all finite Coxeter groups by Bergeron et al. (1992a).
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Index
A-adic topology 17
admissible pyramid 170
algebra of divided powers 108
algebra of functions on the free group 132
alphabet 14
alphabetical order 105
antipode 29
antisymmetry 1
Baker’s identity 36
Bernoulli numbers 72
bialgebra 27
bilinear system 149
binomial coefficient 131
bisection 126
decreasing factorization into Hall words 89
decreasing sequence 86, 137
defect theorem 49
degree 5, 16
dependent 40
derivation 7, 19, 36, 76
derivation tree 91
derived ideal 49
derived series 112
descent 62, 185
descent composition 222
descent set 62, 185
differential polynomial 149
dual coalgebra 38
Dynkins formula 81
Campbell-Baker-Hausdorff formula 56
canonical projections 59
cardinality of a multiset 166
causal analytic functional 148
characteristic 178
Chen series 80
closed 11
circular word 154
code 119
column sum 218
comma-free 122
compatible 227
complete 119
complete symmetric function 202
complete tensor product 28
composition 218
concatenation algebra 15
concatenation product 14, 15
confluent 86
conjugate words 154
conjugation 154, 170
conjugacy class 154, 170
constant term 17
convolution product 28
cycle type 178
cyclic language 160
cyclically standard 174
elementary automorphims 47
empty word 14
enveloping algebra 2
eulerian polynomial 62, 63
evaluation 156, 166
exponent of a word 154
exponential 54
factor 14
factorization of the free monoid 173
finely homogeneous component 16
finely homogeneous polynomial 16
finer 227
finite Lie rank 149
foliage 84
formal series 17
free associative algebra 6, 15
free Lie algebra 4
free Lie p-algebra 35
free Lie superalgebra 102
free magma 4
free monoid 14
free partially commutative Lie algebra 102
free partially commutative monoid 101
free set of Lie polynomials 49
Frobenius image 178
268
Index
generating function 156,179
graded bialgebra 248
Grobner formula 148
Lyndon element 170
Lyndon word 105
Hall polynomial 90
Hall set 84, 89
Hall tree 85
Hall word 89
Hausdorff series 56
highest homogeneous component 43
homogeneous component 16
homogeneous endomorphism 30, 177
homogeneous eulerian polynomial 63
homogeneous polynomial 16
Hopf algebra 29
immediate subtree 84
infiltration 135
infiltration product 134
inner product of symmetric functions 233
inversion 86, 137
invertible 86
iterated integral 56, 148
Jacobi identity 1
jacobian conjecture 51
jacobian matrix 46, 47
Jacobson formulas 33
labelled standard sequence 143
language 38
Lazard set 11
left action 176
left comb 212
left factor 14
left unitary 119
legal rise 86, 137
length 14, 166, 218
letter 14
level 113
Lie algebra 1
Lie bracket 1, 18
Lie dependent 42
Lie endomorphism 30
Lie idempotent 194
Lie invariant 207
Lie monomial 236
Lie polynomial 18
Lie product 1,18
Lie representation 180
Lie series 52
locally finite 18
logarithm 54
lower central series 136
magma 4
Magnus transformation 132
major index 185
message 122
Mobius function 150
multilinear part 178, 180
multiliner polynomial 125
multiplicity in a multiset 166
multiset 166
necklace 154
noncommutative polynomial 15
nontrivial factor 14
nontrivial sesquipower 164
normal basis 171
order associated to a Hall set 114
palindrome 33
partial degree 16
period of a necklace, a word 154
periodic expansion of a word 161
plethysm 202
polynomial 15
power sum 156, 178
prefix 14
primitive element 35, 170
primitive necklace, word 154
product of functions on the free group 132
proper factor 14
proper homomorphism 46
pseudocomposition associated to a
matrix 218
pseudocomposition 218
pyramid 170
quasi-symmetric generating function 245
quasi-symmetric function 242
rank of a free Lie algebra 50
rational series 38
recognizable function 133
recognizable series 38
recognizable subset 150
representative function 133
representative subset 150
restricted enveloping algebra 50
restricted Lie algebra of characteristic p 50
reversal 28
right action 177
Index
269
right factor 14
rise 62, 86, 137
rise set 62
row sum 218
Schur function 186
semi-direct product 8
series 17
sesquipower 164
shift mapping 168
shuffle 24
shuffle algebra 24
shuffle product 24
size 73
space of subword functions 132
standard factorization 89, 170
standard list 126
standard numbering 166
standard permutation 167
standard sequence 85, 136
standard tableau 185
subword 23, 131
subword function 131, 132
subword order 150
substitution of letters 30
suffix 14
suffix code 119
support of the free Lie algebra 33
symmetrized product 57
synchronizing 119
tree 4
type 227
weight 218, 224, 227
weight-preserving 231
word 14
Young subgroup 249
Zassenhaus formula 81
zeta function 174